Question

You are dealt a hand of five cards from a standard deck of 52 playing cards. Calculate the probability of the given type of hand. (None of them is a recognized poker hand.)

(a) Mixed royal meeting: One of each type or royal card (king, queen, jack, ace) of different suites, and a last non-royal card (neither king, queen, jack or ace). Enter a formula. Examples: C(5,3)C(33,3)/C(14,2) C(5,3)C(33,3)C(4,1)/C(100,100) (b) Red and black double royal wedding: A red king, a red queen, a black king, a black queen, and a last non-royal card (neither king, queen, jack or ace).

Enter a formula.
Examples: C(5,3)C(33,3)/C(14,2) C(5,3)C(33,3)C(4,1)/C(100,100)

Answer 1

Five cards from a deck of 52 cards can be selected in 52C5 ways

(a)To select each type of royal card of different suites, initially we select 'king' which can be from any of the 4 suites in 4C1 ways.

Then, we select 'queen' from the remaining 3 suites (as cards should be from different suites, so queen cannot be of the suite from which king is selected) in 3C1 ways

Similarly, 'jack' and 'ace' can be selected in 2C1 and 1C1 ways respectively.

And the last non-royal card can be selected from any of the 36 non royal cards in 36C1 ways.

Thus, required probability = 4C1*3C1*2C1*1C1*36C1/52C5 = 0.000332

(b) Out of each of the 4 cards, there are two cards each of red and black.

Thus, a red king can be selected from two red king in 2C1 ways

Similarly, a red queen can be selected in 2C1 ways

a black king in 2C1 ways

a black queen in 2C1 ways

And the last non royal cards can be selected in 36C1 ways.

Thus, required probability = 2C1*2C1*2C1*2C1*36C1/52C5 = 0.000222