What is the theoretical probability of being dealt exactly three 4's in a 5-card hand from a standard 52-card deck?
To calculate the theoretical probability of being dealt exactly three 4's in a 5-card hand from a standard 52-card deck, we can use the formula for the binomial probability:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the total number of trials (in this case, the total number of ways to choose 5 cards from a 52-card deck, which is 52 choose 5)
- k is the number of successful outcomes (in this case, the number of ways to choose exactly 3 4's from a 4-of-a-kind card in a 52-card deck, which is 4 choose 3).
- p is the probability of success, which is the probability of drawing a 4 from the deck (4/52).
Calculating the binomial probability:
P(X=3) = (52 choose 5) * (4 choose 3) * (4/52)^3 * (48/52)^2
P(X=3) = (2,598,960) * (4) * (64/2704) * (2304/2704)
P(X=3) = 0.0036
Therefore, the theoretical probability of being dealt exactly three 4's in a 5-card hand from a standard 52-card deck is approximately 0.36%.