QUESTION: A friend tosses two coins, and you ask “Is one of them a heads?” The friend replies “Yes.” What is the probability that the other is a heads?
ANSWER: One-third. There are four ways the coins could have landed originally:
- HH: 0.25 probability
- HT 0.25 probability
- TH 0.25 probability
- TT 0.25 probability
(The probability that the first coin lands H is 0.5, and the probability that the second coin lands heads is 0.5, so the probability that each lands heads is 0.5 x 0.5 = 0.25 of HH. Similarly for the others.)
We can rule out TT, since the friend says that one of them landed heads. That leaves 3 possibilities: HH, HT, and TH. Each has equal probability (originally 0.25 each before we got some information, now 0.33 each in light of the information we received).
In two of the cases, the “other” coin is a tails, and in one case it is heads. So the probability that the other coin is heads = ⅓.