Final answer:
To find the probability that at least 5 out of 1,200 callers receive a busy signal, we can use the Poisson approximation to the binomial. The Poisson distribution formula is used to calculate the probabilities. By substituting the values into the formula, we can find the probability that at least 5 out of 1,200 callers received a busy signal.
Step-by-step explanation:
To find the probability that at least 5 out of 1,200 callers receive a busy signal, we can use the Poisson approximation to the binomial. The binomial distribution can be approximated by a Poisson distribution when the number of trials (n) is large and the probability of success (p) is small. In this case, n = 1,200 and p = 0.005. We need to find the probability of x being greater than or equal to 5. Using the Poisson distribution formula, we can calculate:
P(X >= 5) = 1 - P(X < 5)
To calculate P(X < 5) using a Poisson distribution with mean (λ) = np:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Using the formula:
P(X = k) = (e^(-λ) * λ^k) / k!
We can substitute the values into the formula and calculate the probabilities. Finally, we subtract the sum of these probabilities from 1 to get the probability that at least 5 out of 1,200 callers received a busy signal.