PROBLEM 17E

A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of each of the following poker hands:

(a) Four of a kind (four cards of equal face value and one card of a different value).

(b) Full house (one pair and one triple of cards with equal face value).

(c) Three of a kind (three equal face values plus two cards of different values).

(d) Two pairs (two pairs of equal face value plus one card of a different value).

(e) One pair (one pair of equal face value plus three cards of different values).

Answer :

Step 1 of 5 :

Given, A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards.

The claim is to find the probability four of a kind ( four cards of equal face value and one card of a different value.)

The total number of ways different poker hand can be possible =

The number of ways when four of a kind =

=

Therefore, P(four of a kind) =

=

=

= 0.0002400.

Step 2 of 5 :

b)

the claim is to find the probability of full house ( one pair and one triple of cards with equal face value).

Full house ways = ( ways of choosing a pair) ( ways of choosing triple)

= ( ) ( )

= ( 136) (12 4)

= (78) (48)

= 3744

Therefore, P( full house ways) =

=

= 0.00144

The probability of full house ways is 0.00144

Step 3 of 5 :

c)

The claim is to find the probability three of a kind ( three equal face values plus two cards of different values).

3 of a kind = ( )

= ( 134664 4)

= 54912

Therefore, P( 3 of a kind) =

=

= 0.0211

The probability 3 of a kind is 0.0211.