07-10-2017, 01:32 PM
(This post was last modified: 07-10-2017, 02:58 PM by Ebonywood Hellscythe.)
Voted (I'm Dane, voting OM to Ebony)
Alf has 9 coins. Beth also has some coins. The probability that Beth gets more heads when she and Alf flip all their coins is exactly 50%. How many coins does Beth have?
[spoiler]10
Notice that, by symmetry, the probability that Beth gets more tails than Alf is also 50%.
You can also deduce that Beth has more coins than Alf, as the probability that Beth gets the same number of heads (or tails) as Alf must be more than the 50%.
If Alf has more than 1 extra coin, then the p(BethHeads>AlfHeads) + p(BethTails>AlfTails) will be more than 100% as they would not be mutually exclusive and would both be more than 50%. Therefore, Beth only has one more coin.[/spoiler]
Alf has 9 coins. Beth also has some coins. The probability that Beth gets more heads when she and Alf flip all their coins is exactly 50%. How many coins does Beth have?
[spoiler]10
Notice that, by symmetry, the probability that Beth gets more tails than Alf is also 50%.
You can also deduce that Beth has more coins than Alf, as the probability that Beth gets the same number of heads (or tails) as Alf must be more than the 50%.
If Alf has more than 1 extra coin, then the p(BethHeads>AlfHeads) + p(BethTails>AlfTails) will be more than 100% as they would not be mutually exclusive and would both be more than 50%. Therefore, Beth only has one more coin.[/spoiler]