Question

In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9, but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does?

Answer 1

When Mary raises the probability that she has a better hand is 0.273

Explanation:

In the poker game it is given that the probability that Mary has a better hand than John is 0.04.

Let the event that Mary has a better hand than John be A.

Let the event that that Mary raises the stakes be B.

It is also given that if Mary has a better hand than John then she would raise with a probability of 0.9

Therefore p(B | A) = 0.9

If Mary has a poorer hand she will raise with a probability of 0.1

Therefore p(B | A') = 0.1

Given that Mary raises the probability that she has a better hand is given by

p(A | B)

`= (p(A\cap B))/(p(B))` =
`(p(A)p(B | A))/(p(A)p(B | A) + p(A')p(B | A')) = ((0.04 * 0.9) )/((0.04 * 0.9) + (0.96 * 0.1)) = (36)/(36 + 96) = (36)/(132) = (3)/(11)`

= 0.273

So when Mary raises the probability that she has a better hand is 0.273