https://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&feed=atom&action=historyChance News 49 - Revision history2024-03-29T08:21:49ZRevision history for this page on the wikiMediaWiki 1.40.0-alphahttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=22259&oldid=prevBill Peterson: /* Guesstimation */2018-08-29T15:53:12Z<p><span dir="auto"><span class="autocomment">Guesstimation</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>[http://www.nytimes.com/2009/03/31/science/31angi.html?scp=1&sq=biggest%20puzzles&st=cse The biggest of puzzles brought down to size]. <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>[http://www.nytimes.com/2009/03/31/science/31angi.html?scp=1&sq=biggest%20puzzles&st=cse The biggest of puzzles brought down to size]. <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>New York Times, 30 March 2009 <br></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><ins style="font-weight: bold; text-decoration: none;">by Natalie Angier, ''</ins>New York Times<ins style="font-weight: bold; text-decoration: none;">''</ins>, 30 March 2009 <br></div></td></tr>
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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>The article opens by reminding us that with bank bailouts running hundreds of billions of dollars the national debt passing ten trillion, the public need help comparing the magnitudes of really large numbers. For practice, the author recommends so-called &quot;Fermi problems,&quot;. Named for Enrico Fermi, these are estimation problems that physicists and engineers like to use to sharpen their intuition. Two examples cited in the article are: </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 article opens by reminding us that with bank bailouts running hundreds of billions of dollars the national debt passing ten trillion, the public need help comparing the magnitudes of really large numbers. For practice, the author recommends so-called &quot;Fermi problems,&quot;. Named for Enrico Fermi, these are estimation problems that physicists and engineers like to use to sharpen their intuition. Two examples cited in the article are: </div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=15212&oldid=prevBill Peterson: /* Meteorite hits boy */2012-02-20T17:25:01Z<p><span dir="auto"><span class="autocomment">Meteorite hits boy</span></span></p>
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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>[http://www.telegraph.co.uk/scienceandtechnology/science/space/5511619/14-year-old-hit-by-30000-mph-space-meteorite.html "14-year-old hit by 30,000 mph space meteorite"], The Telegraph, June 12, 2009<br></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.telegraph.co.uk/scienceandtechnology/science/space/5511619/14-year-old-hit-by-30000-mph-space-meteorite.html "14-year-old hit by 30,000 mph space meteorite"], <ins style="font-weight: bold; text-decoration: none;">''</ins>The Telegraph<ins style="font-weight: bold; text-decoration: none;">''</ins>, June 12, 2009<br></div></td></tr>
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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>Gerrit Blank survived a direct hit to his hand by a meteorite as it hurtled to Earth at "more than 30,000 miles per hour".<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>Gerrit Blank survived a direct hit to his hand by a meteorite as it hurtled to Earth at "more than 30,000 miles per hour".<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=15211&oldid=prevBill Peterson: /* Meteorite hits boy */2012-02-20T17:24:29Z<p><span dir="auto"><span class="autocomment">Meteorite hits boy</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>From [http://www.wired.com/images_blogs/wiredscience/2009/06/meteorite-nearmisses.jpg Wired magazine], some meteorite "near misses" in history:</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>From [http://www.wired.com/images_blogs/wiredscience/2009/06/meteorite-nearmisses.jpg Wired magazine], some meteorite "near misses" in history:</div></td></tr>
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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"></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>1. How do you think the speed of 30,000 miles per hour was determined?<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>1. How do you think the speed of 30,000 miles per hour was determined?<br> </div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=13289&oldid=prevBill Peterson: /* A new record in craps. */2010-09-30T01:43:17Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
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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>==A new record in craps<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>==A new record in craps==</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>[http://www.time.com/time/nation/article/0,8599,1901663,00.html Holy Craps! How a Gambling Grandma Broke the Record] <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>[http://www.time.com/time/nation/article/0,8599,1901663,00.html Holy Craps! How a Gambling Grandma Broke the Record] <br></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>Time.com, 29 May 2009<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>Time.com, 29 May 2009<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=11211&oldid=prevBill Peterson: /* A new record in craps. */2009-06-25T15:24:26Z<p><span dir="auto"><span class="autocomment">A new record in craps.</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>You can find a discussion of this problem in [http://blogs.wsj.com/numbersguy/crunching-the-numbers-on-a-craps-record-703/ Crunching the numbers on a craps record] from Carl Bialik's Wall Street Journal column, &quot;The Numbers Guy&quot;. He explains why the sevening-out event is complicated, and reports on various attempts to solve the problem. In the final update, he reports that Keith Crank of the American Statistical Association used a Markov chain analysis (which presumably corresponds to what we did above) to find a probability of 1 in 5.6 billion. Earlier in the article he cites simulation results from Professor Michael Shackleford, referencing Shackleford's [http://wizardofodds.com/askthewizard/askcolumns/askthewizard81.html Wizard of Odds] website. Apart from the simulation results, Shackleford presents a recursive computation scheme that is equivalent to the Markov chain. You can also read there interesting historical notes on craps records.</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>You can find a discussion of this problem in [http://blogs.wsj.com/numbersguy/crunching-the-numbers-on-a-craps-record-703/ Crunching the numbers on a craps record] from Carl Bialik's Wall Street Journal column, &quot;The Numbers Guy&quot;. He explains why the sevening-out event is complicated, and reports on various attempts to solve the problem. In the final update, he reports that Keith Crank of the American Statistical Association used a Markov chain analysis (which presumably corresponds to what we did above) to find a probability of 1 in 5.6 billion. Earlier in the article he cites simulation results from Professor Michael Shackleford, referencing Shackleford's [http://wizardofodds.com/askthewizard/askcolumns/askthewizard81.html Wizard of Odds] website. Apart from the simulation results, Shackleford presents a recursive computation scheme that is equivalent to the Markov chain. You can also read there interesting historical notes on craps records.</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>Returning to the Time.com piece, we read &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (<del style="font-weight: bold; text-decoration: none;">for which </del>the expected number of rolls is 6). <del style="font-weight: bold; text-decoration: none;">Going back to </del>the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Returning to the Time.com piece, we read &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the expected number of rolls is 6). <ins style="font-weight: bold; text-decoration: none;">Using </ins>the Markov chain <ins style="font-weight: bold; text-decoration: none;">model</ins>, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a given year.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a given year.</div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=7852&oldid=prevBill Peterson: /* A new record in craps. */2009-06-25T15:23:01Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<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 15:23, 25 June 2009</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l347">Line 347:</td>
<td colspan="2" class="diff-lineno">Line 347:</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 sevened-out state is absorbing; hence the form of row 4. The probabilities for the other rows are easily computed. From state 0, the chance of rolling 2, 3, 7, 11, or 12 is (1+2+6+2+1)/36 = 12/36 which keeps the chain in state 0. The chance of rolling 4 or 10 is (3+3)/36 which leads to state 1; similar calculations hold for transitions to states 2 and 3. Next, from state 1, the chance of reproducing the point is 3/36, which leads to state 0; the chance of rolling a 7 is 6/36 which leads to state 4; otherwise the chain remains in state 1. Rows 2 and 3 are analogous to row 1.</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 sevened-out state is absorbing; hence the form of row 4. The probabilities for the other rows are easily computed. From state 0, the chance of rolling 2, 3, 7, 11, or 12 is (1+2+6+2+1)/36 = 12/36 which keeps the chain in state 0. The chance of rolling 4 or 10 is (3+3)/36 which leads to state 1; similar calculations hold for transitions to states 2 and 3. Next, from state 1, the chance of reproducing the point is 3/36, which leads to state 0; the chance of rolling a 7 is 6/36 which leads to state 4; otherwise the chain remains in state 1. Rows 2 and 3 are analogous to row 1.</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>To analyze the chain, we use standard absorbing chain theory, following notation from Grinstead and Snell's [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to Probability] text. (Concise notes from a recent Dartmouth course are [http://www.math.dartmouth.edu/archive/m20x06/public_html/Lecture14.pdf here]). The leading 4x4 submatrix corresponding to the transient states is denoted <math>Q</math>. The probabilities of still being in particular transient states given by entries in the vector <math>(1,0,0,0)Q^{153}</math>, and the sum of these probabilities is the chance of not having sevened out after 153 rolls. It is easy to implement this computation iteratively, giving <math>1.788824 \times 10^{-10}</math>, which approximately 1 in 5.59 billion. This certainly is a small probability, but much larger than the 1 in 1.56 trillion from Time.com.</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 analyze the chain, we use standard absorbing chain theory, following notation from Grinstead and Snell's [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to Probability] text. (Concise notes from a recent Dartmouth course are [http://www.math.dartmouth.edu/archive/m20x06/public_html/Lecture14.pdf here]). The leading 4x4 submatrix corresponding to the transient states is denoted <math>Q</math>. The probabilities of still being in particular transient states <ins style="font-weight: bold; text-decoration: none;">after 153 rolls is </ins>given by entries in the vector <math>(1,0,0,0)Q^{153}</math>, and the sum of these probabilities is the chance of not having sevened out after 153 rolls. It is easy to implement this computation iteratively, giving <math>1.788824 \times 10^{-10}</math>, which approximately 1 in 5.59 billion. This certainly is a small probability, but much larger than the 1 in 1.56 trillion from Time.com.</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>You can find a discussion of this problem in [http://blogs.wsj.com/numbersguy/crunching-the-numbers-on-a-craps-record-703/ Crunching the numbers on a craps record] from Carl Bialik's Wall Street Journal column, &quot;The Numbers Guy&quot;. He explains why the sevening-out event is complicated, and reports on various attempts to solve the problem. In the final update, he reports that Keith Crank of the American Statistical Association used a Markov chain analysis (which presumably corresponds to what we did above) to find a probability of 1 in 5.6 billion. Earlier in the article he cites simulation results from Professor Michael Shackleford, referencing Shackleford's [http://wizardofodds.com/askthewizard/askcolumns/askthewizard81.html Wizard of Odds] website. Apart from the simulation results, Shackleford presents a recursive computation scheme that is equivalent to the Markov chain. You can also read there interesting historical notes on craps records.</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>You can find a discussion of this problem in [http://blogs.wsj.com/numbersguy/crunching-the-numbers-on-a-craps-record-703/ Crunching the numbers on a craps record] from Carl Bialik's Wall Street Journal column, &quot;The Numbers Guy&quot;. He explains why the sevening-out event is complicated, and reports on various attempts to solve the problem. In the final update, he reports that Keith Crank of the American Statistical Association used a Markov chain analysis (which presumably corresponds to what we did above) to find a probability of 1 in 5.6 billion. Earlier in the article he cites simulation results from Professor Michael Shackleford, referencing Shackleford's [http://wizardofodds.com/askthewizard/askcolumns/askthewizard81.html Wizard of Odds] website. Apart from the simulation results, Shackleford presents a recursive computation scheme that is equivalent to the Markov chain. You can also read there interesting historical notes on craps records.</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>Returning to the Time.com piece, <del style="font-weight: bold; text-decoration: none;">which notes </del>&quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the <del style="font-weight: bold; text-decoration: none;">average </del>number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Returning to the Time.com piece, <ins style="font-weight: bold; text-decoration: none;">we read </ins>&quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (<ins style="font-weight: bold; text-decoration: none;">for which </ins>the <ins style="font-weight: bold; text-decoration: none;">expected </ins>number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a given year.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a given year.</div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=7851&oldid=prevBill Peterson: /* A new record in craps. */2009-06-24T21:59:39Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<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 21:59, 24 June 2009</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l353">Line 353:</td>
<td colspan="2" class="diff-lineno">Line 353:</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>Returning to the Time.com piece, which notes &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the average number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Returning to the Time.com piece, which notes &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the average number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a year.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that a feat like Demauro's would occur in a <ins style="font-weight: bold; text-decoration: none;">given </ins>year.</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 QUESTION:<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 QUESTION:<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=7827&oldid=prevBill Peterson: /* A new record in craps. */2009-06-24T21:58:49Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<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 21:58, 24 June 2009</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l353">Line 353:</td>
<td colspan="2" class="diff-lineno">Line 353:</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>Returning to the Time.com piece, which notes &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the average number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Returning to the Time.com piece, which notes &quot;The average number of dice rolls before sevening out? Eight.&quot; The true value is about 8.5, but it doesn't follow from their original analysis, which asked only how long it takes to roll a seven (the average number of rolls is 6). Going back to the Markov chain, the expected time to absorption is the sum of the row zero entries in the fundamental matrix <math>N = (I-Q)^{-1}</math>. Computing this gives an expected 8.5 rolls to seven-out.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that <del style="font-weight: bold; text-decoration: none;">such an event </del>would occur in a year.</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>Time also interviewed Professor Thomas Cover of Stanford, who pointed out that while the probability of the event in question is small, one needs to remember that there are many people playing craps at any time, all of whom are in principle contending for the record! Shackleford estimates their are about 50 million craps turns per year in the US, giving about a 1% chance that <ins style="font-weight: bold; text-decoration: none;">a feat like Demauro's </ins>would occur in a year.</div></td></tr>
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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>DISCUSSION QUESTION:<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 QUESTION:<br></div></td></tr>
</table>Bill Petersonhttps://www.causeweb.org/wiki/chance/index.php?title=Chance_News_49&diff=7826&oldid=prevBill Peterson: /* A new record in craps. */2009-06-24T21:55:44Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:55, 24 June 2009</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>Claire Suddath</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>Claire Suddath</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><del style="font-weight: bold; text-decoration: none;">The article reports that on </del>May 23, a New Jersey woman named Patricia Demauro set a new world record for the longest turn at craps without &quot;sevening out&quot; by rolling the dice 154 consecutive times. Unfortunately, the <del style="font-weight: bold; text-decoration: none;">lead paragraph </del>misstates the probability calculation needed to compute the odds of her feat<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><ins style="font-weight: bold; text-decoration: none;">On </ins>May 23, a New Jersey woman named Patricia Demauro set a new world record for the longest turn at craps without &quot;sevening out&quot; by rolling the dice 154 consecutive times. Unfortunately, the <ins style="font-weight: bold; text-decoration: none;">article </ins>misstates the probability calculation needed to compute the odds of her feat<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_49&diff=7825&oldid=prevBill Peterson: /* A new record in craps. */2009-06-23T12:59:19Z<p><span dir="auto"><span class="autocomment">A new record in craps.</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:59, 23 June 2009</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;"><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 this illustration, the shooter first establishes a point, namely 5, on the fourth roll. This moves the chain to state 2, where it remains until the shooter reproduces a the 5 on the <del style="font-weight: bold; text-decoration: none;">eight </del>roll. This starts another sequence of come-out attempts. A point of 4 is established on the tenth roll, and the shooter sevens-out on the twelfth roll, ending her turn.</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 this illustration, the shooter first establishes a point, namely 5, on the fourth roll. This moves the chain to state 2, where it remains until the shooter reproduces a the 5 on the <ins style="font-weight: bold; text-decoration: none;">eighth </ins>roll. This starts another sequence of come-out attempts. A point of 4 is established on the tenth roll, and the shooter sevens-out on the twelfth roll, ending her turn.</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>The probability transition matrix ''P'' for the Markov chain is given below:</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 probability transition matrix ''P'' for the Markov chain is given below:</div></td></tr>
</table>Bill Peterson