Final answer:
To calculate the exact probability of making at least three sales out of 80 prospects, you can use the binomial probability formula. To calculate the approximate probability, you can use either the normal approximation or the Poisson approximation to the binomial distribution. The method that produces the most accurate approximation depends on the values of n and p.
Step-by-step explanation:
To calculate the exact probability that the salesperson will make at least three sales out of 80 prospects, we need to use the binomial probability formula. The formula is:
P(X ≥ k) = 1 - P(X < k) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = k-1)]
To calculate the approximate probability using both methods, we can use the normal approximation to the binomial distribution and the Poisson approximation to the binomial distribution. The normal approximation can be used when n is large and p is not close to 0 or 1, while the Poisson approximation can be used when n is large and p is small.
The normal approximation formula is:
P(X ≥ k) ≈ 1 - Φ((k - np) / sqrt(npq))
The Poisson approximation formula is:
P(X ≥ k) ≈ 1 - P(X < k) = 1 - e^(-λ) * (1 + λ/1! + λ^2/2! + ... + λ^(k-1)/(k-1)!)
The method that produces the more accurate approximation depends on the values of n and p. In this specific case, if the probability of a sale on a single contact is 0.15, the normal approximation may be more accurate because the probability of a sale is not small (p = 0.15) and the number of prospects is relatively large (n = 80).