Question

Question 7 (1 point)

Poker game: You pay $5 to play. A 7-card poker hand is dealt, and you are paid $89 if the hand contains a 3 of a kind [at least 3 cards of the same value] and nothing otherwise.

What is the expected value of your payoff from this game?

[Round to 3 digits after decimal point]

Your Answer:

Answer 1

In a 7 card poker game, any 7 cards from the available 52 cards can be dealt.

Total number of ways of selecting 7 cards from 52 = No. of points in our sample space = n(S) = 52C7

Let A be the event that atleast 3 cards of the same value are dealt.

Now, we want to find the number of outcomes favourable to A.

Let us first select the value that is to be repeated 3 times in the hand. This can be selected in 13C1 = 13 ways

Now, the value which is selected has 4 of its kind. Of these we have to select 3. This can be done in 4C3 ways = 4.

Once this is done, the remaining 4 cards can be anything from the remaining 49 cards. They can be selected in 49C4 ways.

Therefore, number of outcomes favourable to A = n(A) = 13×4×49C4

P(A) = n(A)/n(S) = 0.082353

So, we can win $89 with probability 0.082353 and $0 otherwise. Therefore, expected value = 89×0.082353 + 0×(1-0.082353) = $7.329

Expected revenue = 7.329-5 = $2.329