Final answer:
To determine the probability that at most 30 customers have the supermarket club-card in a random sample of 40 customers, we can use the binomial distribution. We need to sum the probabilities of having 0, 1, 2, ..., 30 customers with the club-card using the formula for the probability of getting exactly k successes in n trials. Then, we calculate the probabilities and sum them to find the final probability.
Step-by-step explanation:
To determine the probability that at most 30 customers have the supermarket club-card in a random sample of 40 customers, we can use the binomial distribution. The binomial distribution is used when there are two outcomes (customers with or without the club-card) and a fixed number of trials (40 customers).
To find the probability, we need to sum the probabilities of having 0, 1, 2, ..., 30 customers with the club-card.
The formula for the probability of getting exactly k successes in n trials is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where C(n, k) is the number of combinations of n items taken k at a time, and p is the probability of success (85% or 0.85 in this case).
Let's calculate the probabilities using this formula:
- k = 0: P(X = 0) = C(40, 0) * 0.85^0 * (1-0.85)^(40-0)
- k = 1: P(X = 1) = C(40, 1) * 0.85^1 * (1-0.85)^(40-1)
- ...
- k = 30: P(X = 30) = C(40, 30) * 0.85^30 * (1-0.85)^(40-30)
Finally, we sum all the probabilities to get the probability that at most 30 customers have the supermarket club-card in a random sample of 40 customers.