Question

the worst hand you can get in poker is a 7 high (where you have a 2, 3, 4, 5, 6, 7 and the suits do nit match). is this hand more or less likely than a royal flush (where you have 10, J, Q, K, A and the suits match). Why? this is a 5 card poker game

Answer 1

Royal flush:

A royal flush consists of 10, J, Q, K, A of the same suit.

Now, there are 4 suits of cards. Out of these, one suit can be selected in (₄C₁) or 4 ways. A particular suit has exactly one of each of the cards. 10, J, Q, K, and A. Hence, from the selected suit, the cards 10, J, Q, K, A can be obtained in exactly 1 way.

Without any constraints, the 5 cards in a 5-hand poker game can be selected out of the 52 cards in the deck in (₅₂C₅) or 2,598,960 ways.

The probability of getting a royal flush is:

[(₄C₁) * 1]/(₅₂C₅)

= 4/2,598,960

= 1/649,740

≈ 0.00000154.

Thus, the probability of getting a royal flush is approximately 0.00000154.

7-high:

In a 7 high hand, the possible cards are 2, 3, 4, 5, 6, 7, out of which, any five can be selected. This is possible in (₆C₅) or 6 ways. In the worst hand, the suits of all the five cards must not be the same.

Now, P (suits of all 5 cards are not the same) = 1 – P (suits of all 5 cards are the same).

Out of the 4 suits, one suit can be selected in 4 ways. Then, the probability of selecting five cards from among 2, 3, 4, 5, 6, 7 of the same suit is:

P (suits of all 5 cards are the same)

= (4 * 6)/(₅₂C₅)

= 24/2,598,960

= 1/108,290.

Thus,

P (suits of all 5 cards are not the same)

= 1 – 1/108,290

≈ 0.999990765.

Thus, the probability of getting a 7 high hand such that the suits of all the five cards are not the same is approximately 0.999990765.

Evidently, the probability of a royal flush is much smaller.

Hence, the worst hand is more likely than a royal flush (the best hand).