In a poker game, 5 cards are dealt from a standard 52 card deck that has been we

Question

In a poker game, 5 cards are dealt from a standard 52 card deck that has been well shuffled. You are the only player in this scenario. (Note: if you are not familiar with poker hands, you may want to look up what some of these are.) a) How many different 5 card hands are possible? b) What is the probability that you are dealt two pairs? c) What is the probability that you are dealt a 3 of a kind or 4 of a kind? (Note: 3 of a kind means that the other 5. d) e) 2 cards are NOT a pair.) What is the probability that you are dealt a full house? (Note: a full house means 3 of a kind and a pair) What is the probability that you are dealt a flush, 3 of a kind, or 4 of a kind? (any type of flush is acceptable)

Explanation / Answer

a)52C5=2598960 different 5 card hands are possible.

b)First choose 2 different denominations that will become 2 pairs in 13C2 ways.

For each of these denominations,choose 2 suits from 4 suits available in 4C2*4C2 ways.

The other card and its suit is chosen in 11C1*4C1 ways.

All possible hands can come in 52C5 ways.

Hence ,required probability :

[13C2*4C2*4C2*11C1*4C1]÷52C5=123552/2598960=0.047539015.

c)P(3 of a kind or 4 of a kind)=P(3 of a kind)+ P(4 of a kind).

P(3 of a kind)=

(13C1*4C3*12C2*4C1*4C1)÷52C5.

P(4 of a kind)=

(13C1*4C4*12C1*4C1)÷52C5

P( 3 of a kind or 4 of a kind)=(54912+624)÷2598960=0.021368547.

d)P( Dealt with full house):

(13C1*4C3*12C1*4C2)÷52C5=3744÷2598960=0.001440576

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