Problem of the week #79 - September 30th, 2013

[Jameson]

You are playing some poker with a friend. Say you are dealt 5 cards from a 52 card deck and have three jacks plus the 2 of diamonds and the 5 of hearts. What is the probability that your friend has a higher 3 of a kind, that is three Q's, K's or A's?
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[Jameson]

There were no correct solutions this week.

Solution:

Spoiler

5 cards from the 52 are already accounted for thus our hand is chosen from a pile of 47 cards. For each situation - 3 Kings, Queens or Aces - they all have the same probability so we can find one of them and multiply by 3.

There are 4 Queens from which to choose 3 and 43 cards left over from which to choose the other 2 cards. However these last two cards also cannot be the same, so for the first card we have 43 choices and for the second card we have 39 choices.

\(\displaystyle P[\text{3 Queens}]=\frac{\binom{4}{3} \left(\frac{43 \times 39}{2!}\right)}{\binom{47}{5}} \approx 0.0021865\)

\(\displaystyle P[\text{3 Queens}] \times 3 = \boxed{0.00656}\)