Final answer:
The probability that at least three out of 25 patients calling with claims of having the flu actually have it is calculated using the binomial distribution formula for probability. We subtract the sum of the probabilities of having 0, 1, or 2 patients with the flu from 1 to find the desired probability.
Step-by-step explanation:
The question requires us to calculate the probability that at least three patients out of 25 who claim to have the flu actually do have it, given that the true flu rate is 4%. This is a problem of the binomial distribution where the random variable X represents the number of patients who actually have the flu out of the total claiming to have it. In this case, X can take any value from 0 to 25.
To solve this, we use the formula for the binomial probability:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where C(n, k) is the combination of n items taken k at a time, p is the probability of success on a single trial (having the flu), and n is the number of trials (25 patients).
However, we need the probability of having at least three patients actually have the flu. This means we sum the probabilities from three to the maximum number possible, which is 25:
P(X ≥ 3) = 1 - (P(X < 3))
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Calculating these probabilities using the binomial formula and then summing them will give us the probability of having fewer than three patients with the flu. Subtracting this value from 1 gives us the probability that at least three out of the 25 patients actually have the flu.