https://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&feed=atom&action=historyChance News 80 - Revision history2024-03-19T10:35:33ZRevision history for this page on the wikiMediaWiki 1.40.0-alphahttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=15907&oldid=prevBill Peterson: /* Marilyn slips up on a drug testing question */2012-05-09T19:09:17Z<p><span dir="auto"><span class="autocomment">Marilyn slips up on a drug testing question</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:by Marilyn vos Savant, ''Parade'', 2 January 2012</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:by Marilyn vos Savant, ''Parade'', 2 January 2012</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Here Marilyn suggests that the original problem may have been ambiguous. Nevertheless, she does print corrections from Jerry and another reader. She also swears off eggnog, and promises a followup on January 22. Check back here for the update<del style="font-weight: bold; text-decoration: none;">!</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Here Marilyn suggests that the original problem may have been ambiguous. Nevertheless, she does print corrections from Jerry and another reader. She also swears off eggnog, and promises a followup on January 22. Check back here for the <ins style="font-weight: bold; text-decoration: none;">[http://test.causeweb.org/wiki/chance/index.php/Chance_News_81#Marilyn.27s_correction_on_the_drug-testing_problem </ins>update<ins style="font-weight: bold; text-decoration: none;">].</ins></div></td></tr>
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</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=15068&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-02-08T20:30:49Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. According to the article, his great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. According to the article, his great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, <del style="font-weight: bold; text-decoration: none;">the figure reported </del>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 ''or more'' successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, <ins style="font-weight: bold; text-decoration: none;">this </ins>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 ''or more'' successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
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</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=15067&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-02-08T20:29:21Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. According to the article, his great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. According to the article, his great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, the figure <del style="font-weight: bold; text-decoration: none;">given </del>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 ''or more'' successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, the figure <ins style="font-weight: bold; text-decoration: none;">reported </ins>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 ''or more'' successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=15066&oldid=prevBill Peterson: /* Winning the fight against crime by putting your head in the sand */2012-02-08T20:28:09Z<p><span dir="auto"><span class="autocomment">Winning the fight against crime by putting your head in the sand</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Winning the fight against crime by putting your head in the sand==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Winning the fight against crime by putting your head in the sand==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[http://www.nytimes.com/2011/12/31/nyregion/nypd-leaves-offenses-unrecorded-to-keep-crime-rates-down.html Police Tactic: Keeping Crime Reports Off the Books] Al Baker and Joseph Goldstein, <del style="font-weight: bold; text-decoration: none;">The </del>New York Times, December 30, 2011.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[http://www.nytimes.com/2011/12/31/nyregion/nypd-leaves-offenses-unrecorded-to-keep-crime-rates-down.html Police Tactic: Keeping Crime Reports Off the Books] Al Baker and Joseph Goldstein, <ins style="font-weight: bold; text-decoration: none;">''</ins>New York Times<ins style="font-weight: bold; text-decoration: none;">''</ins>, December 30, 2011.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Police officers are joining just about every other profession in trying to skew the statistics to make themselves look good.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Police officers are joining just about every other profession in trying to skew the statistics to make themselves look good.</div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=15065&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-02-08T20:25:54Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. <del style="font-weight: bold; text-decoration: none;">His </del>great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. <ins style="font-weight: bold; text-decoration: none;">According to the article, his </ins>great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, <del style="font-weight: bold; text-decoration: none;">this </del>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 or more successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, <ins style="font-weight: bold; text-decoration: none;">the figure given </ins>is the binomial probability for obtaining exactly 56 successes (gimels) in 68 trials (spins) with success probability 1/4; this assumes independent spins of a perfectly balanced dreidel. But there are two problems here. First, we should ask for the chance of 56 <ins style="font-weight: bold; text-decoration: none;">''</ins>or more<ins style="font-weight: bold; text-decoration: none;">'' </ins>successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=14904&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-01-18T16:01:58Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:01, 18 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l189">Line 189:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 <del style="font-weight: bold; text-decoration: none;">(assuming </del>independent spins of a perfectly balanced dreidel<del style="font-weight: bold; text-decoration: none;">)</del>. <del style="font-weight: bold; text-decoration: none;">There </del>are two problems here. First, we should ask for the chance of 56 or more successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes <ins style="font-weight: bold; text-decoration: none;">(gimels) </ins>in 68 trials <ins style="font-weight: bold; text-decoration: none;">(spins) </ins>with success probability 1/4<ins style="font-weight: bold; text-decoration: none;">; this assumes </ins>independent spins of a perfectly balanced dreidel. <ins style="font-weight: bold; text-decoration: none;">But there </ins>are two problems here. First, we should ask for the chance of 56 or more successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include any shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=14903&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-01-18T15:59:34Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:59, 18 January 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This adjustment does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, the binomial description is not correct because it allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include <ins style="font-weight: bold; text-decoration: none;">any </ins>shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In an effort to describe the order of magnitude of the answer, the article reports that the figure was 22.5 billion times 1 trillion. Do you think this helps the lay person to understand it? Can you suggest an alternative?</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In an effort to describe the order of magnitude of the <ins style="font-weight: bold; text-decoration: none;">(originally reported) </ins>answer, the article reports that the figure was 22.5 billion times 1 trillion. Do you think this helps the lay person to understand it? Can you suggest an alternative?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Submitted by Bill Peterson, based on a suggestion from Adam Peterson</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Submitted by Bill Peterson, based on a suggestion from Adam Peterson</div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=14902&oldid=prevBill Peterson at 01:35, 18 January 20122012-01-18T01:35:18Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:35, 18 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">December 23, 2011 to January 17, 2012</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Quotations==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Quotations==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>“The role of context. .... The focus on variability naturally gives statistics a particular content that sets it apart from from mathematics itself and from other mathematical sciences, but there is more than just content that distinguishes statistical thinking from mathematics. Statistics requires a different kind of thinking, because <i>data are not just numbers, they are numbers with a context</i>. .... <i>In mathematics, context obscures structure. .... In data analysis, context provides meaning.</i> .... [A]lthough statistics cannot prosper without mathematics, the converse fails.”<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>“The role of context. .... The focus on variability naturally gives statistics a particular content that sets it apart from from mathematics itself and from other mathematical sciences, but there is more than just content that distinguishes statistical thinking from mathematics. Statistics requires a different kind of thinking, because <i>data are not just numbers, they are numbers with a context</i>. .... <i>In mathematics, context obscures structure. .... In data analysis, context provides meaning.</i> .... [A]lthough statistics cannot prosper without mathematics, the converse fails.”<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=14896&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-01-18T01:27:40Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:27, 18 January 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This <del style="font-weight: bold; text-decoration: none;">correction </del>does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However,<del style="font-weight: bold; text-decoration: none;">observe that </del>the binomial description allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This <ins style="font-weight: bold; text-decoration: none;">adjustment </ins>does change the order of magnitude of the probability, which becomes 1 in <math>2.09 \times 10^{22}</math>. However, <ins style="font-weight: bold; text-decoration: none;"> </ins>the binomial description <ins style="font-weight: bold; text-decoration: none;">is not correct because it </ins>allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_80&diff=14895&oldid=prevBill Peterson: /* Surprising dreidel outcome */2012-01-18T01:26:52Z<p><span dir="auto"><span class="autocomment">Surprising dreidel outcome</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:26, 18 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l187">Line 187:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To the amazement everyone present, Alfred Lorini compiled a streak of 68 spins that included 56 gimels and zero shins. His great-nephew "used a binomial distribution and came up with 1-in-2.25 times 10 to the 22nd power for the order of magnitude."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This correction does change the order of magnitude of the <del style="font-weight: bold; text-decoration: none;">answer</del>. However,observe that the binomial description allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In fact, this is the binomial probability for obtaining exactly 56 successes in 68 trials with success probability 1/4 (assuming independent spins of a perfectly balanced dreidel). There are two problems here. First, we should ask for the chance of 56 or more successes. This correction does change the order of magnitude of the <ins style="font-weight: bold; text-decoration: none;">probability, which becomes 1 in <math>2.09 \times 10^{22}</math></ins>. However,observe that the binomial description allows the non-gimel rolls to be nun, hei or shin. Mr. Lorini's feat was more unusual in that the non-gimels did not include shins. Thus we really need to consider a multinomial situation, with categories (gimel, shin, neither), for which the probabilities are (1/4, 1/4, 1/2). The chance of 56 or more gimels in 68 rolls, with zero shins, is then calculated as 1 in <math>2.62 \times 10^{24}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Discussion'''<br></div></td></tr>
</table>Bill Peterson