Final answer:
The student's question involves probability distributions, specifically the exponential and uniform distributions. The detailed answers provide explanations for calculating the probabilities of various events related to website visit durations and intervals between bus arrivals.
Step-by-step explanation:
The student's question appears to involve understanding probability distributions and their applications, particularly the exponential distribution and the uniform distribution. We will break down each part of the question to provide clear answers using the provided information.
a. Probability of Duration between Visits Being More Than 10 Minutes
With the rate of 12 visits per hour, the average time between visits is 5 minutes (since 60 minutes / 12 visits = 5 minutes). The exponential distribution function is given by F(t) = 1 - e^(-λt), where λ is the rate parameter (here, 12 visits per hour or 0.2 visits per minute), t is the time, and e is the base of the natural logarithm. To find the probability that the duration between two successive visits is more than 10 minutes, we calculate P(T > 10) = 1 - F(10).
b. Top 25 Percent of Durations between Visits
To find the duration that corresponds to the top 25 percent of durations between visits, we need to find the 75th percentile of the exponential distribution. This involves solving F(t) = 0.75 for t, which gives us the value at which the top 25 percent of durations are at least that long.
c. Probability of Next Visit Occurring Within 5 Minutes After 20 Minutes
If 20 minutes have passed since the last visit, the exponential distribution's memoryless property ensures that the probability of the next visit occurring within the next 5 minutes is the same as if no time had passed. Therefore, we calculate P(T ≤ 5).
d. Probability of Fewer than Seven Visits in One Hour
This can be answered using the Poisson distribution, where we find the probability of having 0 to 6 visits in a given hour with a mean rate of 12 visits per hour.
The rest of the examples provided, such as those involving waiting for a bus, are unrelated to the original question about a browser clickpath.