Final answer:
To find the probability of selecting exactly three clubs from a standard deck, where the card is replaced after each selection and the experiment is repeated five times, we use the binomial probability formula. The probability is approximately 0.0404, or 4.04%.
Step-by-step explanation:
To find the probability of selecting exactly three clubs from a standard deck, where the card is replaced after each selection and the experiment is repeated five times, we can use the binomial probability formula:
P(X=k) = C(n, k) * p^k * q^(n-k)
where:
- X is the number of successes (selecting clubs) in n trials
- P(X=k) is the probability of getting k successes
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on any single trial (the probability of selecting a club from a standard deck is 13/52)
- q is the probability of failure on any single trial (the probability of not selecting a club, i.e., selecting a non-club card, is 39/52)
- n is the number of trials (in this case, 5)
- k is the number of successes we want (in this case, 3)
Plugging the values into the formula, we have:
P(X=3) = C(5, 3) * (13/52)^3 * (39/52)^2
Simplifying the expression and calculating, we find that the probability of selecting exactly three clubs is approximately 0.0404, or 4.04%.