Question

A pinochle deck consists of two copies each of the 9, 10, J, Q, K and A cards of all 4 suits, for a total of 48 cards. You draw 5 cards at random from a pinochle deck

a. what is the probability of drawing a flush ( all 5 cards are the same suit)?

b. What is the probability of drawing a full house ( 3 cards of one value, 2 cards of a different value)?

(Use combinational probability) nCn

Answer 1

There are total 48 cards. 5 cards drawn at random form pinochle deck.

That is have to choose 5 from 48, use combination to find the total ways of selecting 5 cards

There are 4 suits of each value.

a)

Probability of drawing Royal flush.

All 5 cards are the same suit.

There are 4 suits so select 1 suit from 4 first, for 6 value cards there are each 2 of same suit since Pinochle deck consist two copies

So select 5 from 12. There are 2 repeated cards so subtract the straight flush from this to get the total flush cards.

So ways to select flush cards is

Since two cards of each value are repeated,

= 3168 - 256 = 2912

b) Probability of drawing a full house

3 cards of same value.

So first select 1 values type from the 6 (9, 10, J, Q, K and A), then select 3 cards from that 8, since there are 8 cards for each value. That is

Now select two different value cards from the remaining 5 and then select 1 from each 8. Because in each there are 8 cards.

That is

Total ways to select Full house are