The number of calls received by a car towing service averages 21.6 per day (per 24-hour

period). Find the probability that in a randomly selected hour the number of calls is fewer than 4. What
requirement for the Poisson distribution is probably not valid in the scenario outlined?

To find the probability that in a randomly selected hour the number of calls is fewer than 4, we can use the Poisson distribution formula. The formula for the Poisson distribution is:

P(x; λ) = (e^(-λ) * λ^x) / x!

Where:
P(x; λ) is the probability of getting exactly x events in a given interval
e is the base of the natural logarithm (approximately 2.71828)
λ is the average number of events in the interval

In this case, we know that the average number of calls in a 24-hour period is 21.6. Therefore, the average number of calls in a 1-hour period (λ) can be calculated by dividing the average number of calls in a 24-hour period by 24:

λ = 21.6 / 24 = 0.9

Now, we need to calculate the probability of getting fewer than 4 calls in a random hour, so we need to calculate P(x < 4; 0.9). We can do this by summing the probabilities of getting exactly 0, 1, 2, and 3 calls:

P(x < 4; 0.9) = P(x = 0; 0.9) + P(x = 1; 0.9) + P(x = 2; 0.9) + P(x = 3; 0.9)

Let's calculate these probabilities:

P(x = 0; 0.9) = (e^(-0.9) * 0.9^0) / 0! = e^(-0.9) ≈ 0.406

P(x = 1; 0.9) = (e^(-0.9) * 0.9^1) / 1! = 0.9 * e^(-0.9) ≈ 0.365

P(x = 2; 0.9) = (e^(-0.9) * 0.9^2) / 2! = 0.405 * e^(-0.9) ≈ 0.182

P(x = 3; 0.9) = (e^(-0.9) * 0.9^3) / 3! = 0.27 * e^(-0.9) ≈ 0.061

Now, we can sum these probabilities:

P(x < 4; 0.9) = 0.406 + 0.365 + 0.182 + 0.061 ≈ 0.903

Therefore, the probability that in a randomly selected hour the number of calls is fewer than 4 is approximately 0.903, or 90.3%.

Now, let's identify the requirement for the Poisson distribution that is probably not valid in the scenario outlined. The Poisson distribution assumes that the number of events in non-overlapping intervals is independent. In the case of a car towing service, it is possible that the number of calls in consecutive hours may be influenced by the previous hour. For example, if there were a significant number of calls in one hour, it may lead to more calls in the following hour due to delays or other factors. Therefore, the assumption of independence may not be valid in this scenario.