Question

A 5-card hand is dealt from a perfectly shuffled deck of playing cards. What is the probability that the hand is a full house?

Discrete math
ANSWERED
asked 2021-08-03

A 5-card hand is dealt from a perfectly shuffled deck of playing cards. What is the probability that the hand is a full house? A full house has three cards of the same rank and another pair of the same rank. For example, \(\{4\spadesuit,\ 4\heartsuit,\ 4\diamondsuit,\ J\spadesuit,\ J\clubsuit\}\)

Answers (1)

2021-08-04
Step 1
Total number of cards \(\displaystyle={52}\)
Total number of suits \(\displaystyle={4}\)
Number of cards in each suit \(\displaystyle={13}\)
number of ranks in each suit \(\displaystyle={13}\)
The probability that the hand is a full house is given as:
Full house is having three cards of same rank and another pair of the same rank
P(hand is a full house) \(\displaystyle={\frac{{{13}{C}_{{{1}}}\times{4}{C}_{{{3}}}\times{12}{C}_{{{1}}}\times{4}{C}_{{{2}}}}}{{{52}{C}_{{{5}}}}}}\)
\(\displaystyle={\frac{{{13}\times{4}\times{12}\times{6}}}{{{2598960}}}}\)
\(\displaystyle={\frac{{{3744}}}{{{2598960}}}}\)
\(\displaystyle={0.00144}\)
Step 2
Therefore, the probability that the hand is a full house is \(\displaystyle{0.00144}\)
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