A coin having probability .8 of landing on heads is flipped. A observes the result—either heads or tails—and rushes off to tell B. However, with probability .4, A will have forgotten the result by the time he reaches B. If A has forgotten, then, rather than admitting this to B, he is equally likely to tell B that the coin landed on heads or that it landed tails. (If he does remember, then he tells B the correct result.) (a) What is the probability that B is told that the coin landed on heads? (b) What is the probability that B is told the correct result? (c) Given that B is told that the coin landed on heads, what is the probability that it did in fact land on heads?

The probability that the coin lands on heads and A does not forget is: 0.8 x 0.6 = 0.48
The probability that the coin lands on tails and A does not forget is similarly: 0.2 x 0.6 = 0.12
The probability of A forgetting the result and telling B it's heads is: 0.4 x 0.5 = 0.2 (This probability is the same as when A forgets and tells B it's tails too.)

Using these results lets answer parts (a) through (c):

a) The probability that B is told it landed on heads is going to be 0.48 + 0.2. This takes into account scenario 1, and 3 where A tells B the result is heads.

b) The probability that B is told the correct answer is 0.48 + 0.12 + (the probability A is correct when telling B the result after forgetting). For this, we need to consider scenario 3, and multiply it with the probability of being right.

For heads, this is: 0.2 x 0.8 = 0.16

For tails, this is: 0.2 x 0.2 = 0.04

Finally, the probability B is told the correct answer is:

0.48 + 0.12 + 0.16 + 0.04 = 0.8

c) Since it is stated that B is told the coin landed on heads, we do not need to consider whether A is telling the truth or not. We only need to state the probability of getting heads when the coin is flipped. Thus this answer is 0.8.