tag:blogger.com,1999:blog-15628310.post257669989574886535..comments2022-07-26T20:48:08.865-04:00Comments on Question of the day: Avoid the Flooding TodayAnonymoushttp://www.blogger.com/profile/18153935609499338685noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-15628310.post-35277146676357995942010-03-18T13:56:21.763-04:002010-03-18T13:56:21.763-04:00Thank you for the interesting puzzle, which genera...Thank you for the interesting puzzle, which generated quite a bit of thought in my maths department, as well as three solutions of various levels of complexity. <br /><br />I don't think I can include pictures on the answers here, so see <br /><br />http://joningram.org/blog/2010/03/one-more-coin-three-solutions/<br /><br />for the low level one, but the high-level argument is very nice:<br /><br />Let p be the probability that A throws more heads than B.<br /><br />1 - p = P(A throws the same or fewer heads than B) = P(A throws more tails than B) = P(A throws more heads than B) = p.<br /><br />So 2p = 1, and the probability of A throwing more heads than B is exactly 1/2.Jon Ingramhttp://joningram.orgnoreply@blogger.comtag:blogger.com,1999:blog-15628310.post-21265145030129183562010-03-16T13:10:32.210-04:002010-03-16T13:10:32.210-04:00Gee, probability is never intuitive to me but that...Gee, probability is never intuitive to me but that was one of the few problems that was. The other coins don't enter into the solution, only PlayerA's extra coin, which has 50% chance of landing heads.Anonymoushttps://www.blogger.com/profile/07495207276865738681noreply@blogger.comtag:blogger.com,1999:blog-15628310.post-15807990743838183112010-03-16T10:58:15.110-04:002010-03-16T10:58:15.110-04:00I'm not surprised to see you got it, Andy. Th...I'm not surprised to see you got it, Andy. The chances are 50/50, although the answer is not intuitive at all.Anonymoushttps://www.blogger.com/profile/18153935609499338685noreply@blogger.comtag:blogger.com,1999:blog-15628310.post-60533297742595634372010-03-15T11:06:30.936-04:002010-03-15T11:06:30.936-04:0050%
If you go case by case, you can see the patte...50%<br /><br />If you go case by case, you can see the pattern.<br /><br />If B has 0 coins and A has 1:<br />A has a 1/2 chance of getting a head, and therefore having more heads than B.<br /><br />If B has 1 and A has 2:<br /><br />A has a 1/4 chance of having 2 heads (which will always beat B), and 1/2 chance of having 1 head (which will beat B if he throws tails).<br />B has a 1/2 chance of throwing tails.<br />So putting that together, the chances of A having more heads than B:<br /><br />1/4 + (1/2 * 1/2) = 1/2<br /><br />If B has 2 coins and A has 3:<br /><br />1/8 chance A throws 3 heads<br />3/8 chance A throws 2 heads<br />3/8 chance A throws 1 head<br /><br />1/4 chance B throws no heads<br />1/2 chance B throws 1 head<br /><br />So the chances of A throwing more heads than B =<br /><br />1/8 + (3/8 * 3/4) + (3/8 * 1/4)<br /><br />4/32 + 9/32 + 3/32 = 1/2Andyhttps://www.blogger.com/profile/14117563810484999900noreply@blogger.com