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

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, $$\{4\spadesuit,\ 4\heartsuit,\ 4\diamondsuit,\ J\spadesuit,\ J\clubsuit\}$$

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 $$\displaystyle{0.00144}$$