Sandbox: Difference between revisions
| Line 9: | Line 9: | ||
Continues "And here is my recipe for getting the most out of this problem: if you can solve it, do not stop with one answer. Rather, see how many answers you can come up with. I’ve posed this problem to many people, and I continue to hear novel solutions." | Continues "And here is my recipe for getting the most out of this problem: if you can solve it, do not stop with one answer. Rather, see how many answers you can come up with. I’ve posed this problem to many people, and I continue to hear novel solutions." | ||
Here are three ways that suggested themselves quickly: | |||
* Toss the coin until the first head appears. You win if this takes an even number of tosses | |||
* Toss the coin twice. You win on HH and lose on HT or TH. If TT appears, ignore the result and make another two tosses. | |||
* Toss the coin until the first appearance of HTT or HHT on consecutive tosses. You win HTT. | |||
The third is an instance of the game [http://en.wikipedia.org/wiki/Penney's_game Penney-Ante], invented by William Penney. It is a famous example of non-transitivity: whatever triple you choose, I can choose one that has a better than even chance of coming up first. What I did not know until I searched for this online description, was that there is a known variation with cards, called the [http://en.wikipedia.org/wiki/Penney's_game#Variation_with_playing_cards Humble-Nishiyama Randomness Game] | |||
Submitted by Bill Peterson | Submitted by Bill Peterson | ||
Revision as of 01:44, 7 April 2014
A coin puzzle
A coin problem
by Gary Antonik, Numberplay blog, New York Times, 17 March 2014
The post begins with this simple problem, posed by Daniel Finkel:
Consider this simple game: flip a fair coin twice. You win if you get two heads, and lose otherwise. It’s not hard to calculate that the chances of winning are 1/4… . Your challenge is to design a game, using only a fair coin, that you have a 1/3 chance of winning.
Continues "And here is my recipe for getting the most out of this problem: if you can solve it, do not stop with one answer. Rather, see how many answers you can come up with. I’ve posed this problem to many people, and I continue to hear novel solutions."
Here are three ways that suggested themselves quickly:
- Toss the coin until the first head appears. You win if this takes an even number of tosses
- Toss the coin twice. You win on HH and lose on HT or TH. If TT appears, ignore the result and make another two tosses.
- Toss the coin until the first appearance of HTT or HHT on consecutive tosses. You win HTT.
The third is an instance of the game Penney-Ante, invented by William Penney. It is a famous example of non-transitivity: whatever triple you choose, I can choose one that has a better than even chance of coming up first. What I did not know until I searched for this online description, was that there is a known variation with cards, called the Humble-Nishiyama Randomness Game
Submitted by Bill Peterson