Final answer:
The probability that no event occurs between 8 p.m. and 9 p.m. is approximately 0.1353. The expected time for the fourth event to occur is 2 hours. The probability that two or more events occur between 6 p.m. and 8 p.m. is approximately 0.594.
Step-by-step explanation:
(a) To find the probability that no event occurs between 8 p.m. and 9 p.m., we need to calculate the average number of events during that time period. Since the rate of events is 2 per hour, the average number of events in a 1-hour period is 2. Therefore, the probability of no events occurring in a one hour period is:
P(no event) = e^(-2) ≈ 0.1353
(b) To find the expected time at which the fourth event occurs, we need to calculate the average time between events. Since the rate is 2 per hour, the average time between events is 1/2 hour or 30 minutes. Therefore, the expected time for the fourth event to occur is:
Expected time = 30 minutes * 4 = 120 minutes or 2 hours
(c) To find the probability that two or more events occur between 6 p.m. and 8 p.m., we need to calculate the probability of fewer than two events occurring. Using the Poisson distribution, the probability of exactly one event occurring is
P(1 event) = e^(-2) * 2^1 / 1! ≈ 0.2707
Therefore, the probability of two or more events occurring is:
P(≥2 events) = 1 - P(0 event) - P(1 event) ≈ 0.594