Probability

[chelsea88]

Find the probability of the following card hands from a 52 card deck. In poker, aces are either high or low. A bridge hand is made up of 13 cards.
1) In bridge, exactly 3 kings and exactly 3 queens.

 

[soroban]

Hello, chelsea88!

Find the probability of the following card hands from a 52-card deck.
A bridge hand is made up of 13 cards.

1) Exactly 3 Kings and exactly 3 Queens.

\(\displaystyle \text{There are: }\:{52\choose13}\,\text{ possible bridge hands.}\)


\(\displaystyle \text{In the deck, there are: }\,\text{4 Kings, 4 Queens, and 44 Others.}\)

\(\displaystyle \text{We want: }\;\begin{Bmatrix}\text{3 of the 4 Kings: } & {4\choose3}\text{ ways} \\ \\[-3mm] \text{3 of the 4 Queens: } & {4\choose3}\text{ ways} \\ \\[-3mm] \text{7 of the 44 Others:} & {44\choose7}\text{ ways} \end{Bmatrix}\)

\(\displaystyle \text{Hence, there are: }\:{4\choose3}{4\choose3}{44\choose7}\text{ ways to get 3 Kings, 3 Queens, and 7 Others.}\)


\(\displaystyle \text{Therefore: }\;P(\text{3 Kings, 3 queens, 7 Others}) \;=\;\frac{{4\choose3}{4\choose3}{44\choose7}}{{52\choose13}}\)

I'll let you crank it out . . .

 

[chelsea88]

Thank you so much for your reply, you explained everything very clearly. But I think I am not doing something correctly when I solve it. When I multiple (4/3)(4/3)(44/7) I get 11.17 or 704/63. Then when I divide this by 52/13, I get 2.8. Shouldn't the number I'm getting be a decimal?
Thank you!

 

[galactus]

Those are combinations, not fractions.

i.e. \(\displaystyle \binom{4}{3}=\frac{4!}{3!(4-3)!}=4\)

 

[chelsea88]

Oh, that makes much more sense. So when I solve this equation I get 4.83 E-4, which is equivalent to .00048 correct?

 

[galactus]

Not quite. It is twice that. .000965....

 

[chelsea88]

Oh, I just found my math error, now I got it. Thanks so much