# Find the probability questions

• Dec 6th 2006, 03:17 PM
jenneedshelp
Find the probability questions
Hey I was wondering if anyone could help me out with these two questions:

Find the probability of:

1) a five-card poker hand dealt from a standard deck of 52 playing cards results in full house (3 of a kind, 2 of a kind)

2) Two face cards are drawnin a row (not replaced) from a standard deck of 52 playing cards given that the first card drawn is a king.

• Dec 6th 2006, 04:31 PM
Soroban
Hello, jenneedshelp!

Quote:

Find the probability of:

1) a five-card poker hand dealt from a standard deck of cards
results in full house (3 of a kind, 2 of a kind)

There are: .$\displaystyle \binom{52}{5}\:=\:2,598,960$ possible 5-card hands.

For the Triple, there are $\displaystyle 13$ choices for its value.
. . There are $\displaystyle \binom{4}{3} = 4$ ways to select the Triple.

For the Pair, there are $\displaystyle 12$ choices for its value.
. . There are $\displaystyle \binom{4}{2} = 6$ ways to select the Pair.

Hence, there are: .$\displaystyle 13\cdot4\cdot12\cdot6\:=\:3744$ possible Full Houses.

Therefore: .$\displaystyle P(\text{Full House})\;=\;\frac{3774}{2,598,960} \;=\;\frac{6}{4165}$

Quote:

2) Two face cards are drawn in a row (not replaced) from a standard deck of cards,
given that the first card drawn is a king.

We don't need Bayes' Theorem or anything fancy for this one . . .

There are $\displaystyle 12$ Face Cards in a standard deck of cards.

If the first card drawn is a King,
. . there are $\displaystyle 11$ Face Cards left among the remaiing $\displaystyle 51$ cards.

Therefore: .$\displaystyle P(\text{2nd is Face}|\text{1st is a King}) \:=\:\frac{11}{51}$

• Dec 6th 2006, 04:39 PM
galactus
[quote]Find the probability of:

1) a five-card poker hand dealt from a standard deck of 52 playing cards results in full house (3 of a kind, 2 of a kind)
Quote:

3 out of 4 suits: C(4,3)
2 out of 4 suits: C(4,2)
2 out of 13 different face values(order matters): P(13,2)
5 cards out of 52: C(52,5)

Therefore, $\displaystyle \frac{P(13,2)C(4,3)C(4,2)}{C(52,5)}=\frac{3744}{25 98960}=\frac{6}{4165}\approx{0.00144}$

Quote:

2) Two face cards are drawnin a row (not replaced) from a standard deck of 52 playing cards given that the first card drawn is a king.
If the first is a king, then there are 11 face cards left, therefore,
11/51