According to an estimate, 50% of the people in America have at least one credit card. If a random sample of 30 persons is selected, what is the probability that at most 18 of them will have at least one credit card?
To find the probability that at most 18 out of 30 people will have at least one credit card, we need to use the binomial probability formula. The binomial probability formula is:
P(X = k) = C(n, k) * p^k * q^(n-k),
where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of ways of choosing k successes from n trials (also known as the binomial coefficient),
- p is the probability of success on a single trial, and
- q is the probability of failure on a single trial.
In this case, the probability of success is the proportion of people in America who have at least one credit card, which is 50% or 0.5. Therefore, p = 0.5.
The probability of failure, q, is 1 - p. So, q = 1 - 0.5 = 0.5.
To find the probability of at most 18 people having at least one credit card, we sum up the probabilities of having 0, 1, 2, ..., 18 successes:
P(X ≤ 18) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 18)
Using the formula mentioned earlier, we can calculate each of these probabilities and sum them up to get the final answer.