An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of 6.4 emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution.
(a)
What is the probability Linda Lahey, company president, received exactly 1 email between 4 P.M. and 5 P.M. yesterday? (Round your answer to 4 decimal places.)
(b)
What is the probability she received 8 or more emails during the same period? (Round your answer to 4 decimal places.)
(c)
What is the probability she received four or less emails during the period? (Round your answer to 4 decimal places.)
To determine the probabilities in this problem, we can use the Poisson distribution formula. The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, given the average rate at which these events occur.
The formula for the Poisson distribution is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!,
where x is the number of events, λ is the average rate of events, e is the base of the natural logarithm (approximately 2.71828), and x! is the factorial of x.
(a) To find the probability that Linda received exactly 1 email between 4 P.M. and 5 P.M. yesterday, we need to calculate P(x=1; λ), where λ is the average rate of 6.4 emails per hour. Substitute these values into the formula:
P(x=1; 6.4) = (e^(-6.4) * 6.4^1) / 1!
Calculating this expression will give you the answer to part (a).
(b) To find the probability that Linda received 8 or more emails between 4 P.M. and 5 P.M. yesterday, we need to calculate the cumulative probability for x ≥ 8. This involves summing up the probabilities for each value of x from 8 to infinity:
P(x ≥ 8; 6.4) = P(x=8; 6.4) + P(x=9; 6.4) + ...
Since summing an infinite number of probabilities is not feasible, we need to approximate the value by summing a sufficient number of terms. In this case, we can sum up the probabilities from x=8 to x=20, for example, until the sum converges to a reasonable approximation.
Calculate each individual probability using the Poisson formula and sum them up until the approximation is accurate enough.
(c) To find the probability that Linda received four or less emails between 4 P.M. and 5 P.M. yesterday, we need to calculate the cumulative probability for x ≤ 4. This involves summing up the probabilities for each value of x from 0 to 4:
P(x ≤ 4; 6.4) = P(x=0; 6.4) + P(x=1; 6.4) + ... + P(x=4; 6.4)
Calculate each individual probability using the Poisson formula and sum them up to get the answer to part (c).
Remember to round your answers to 4 decimal places as requested.