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Let's break down how to calculate probabilities using the Poisson distribution. In this problem, hits to a website happen at the rate of 1.4 per minute. We want to find the probability of different numbers of hits within a specific time.

The first step is identifying the rate of hits (\textbackslash\textlambda), which is 1.4 hits per minute. Our time interval is from 7:30 p.m. to 7:35 p.m., which is 5 minutes.

So, we calculate the expected number of hits, \textbackslash\textlambda, as:

\textbackslash\textlambda = 1.4 \textbackslash times 5 = 7.

To find the probability of exactly seven hits, we use the Poisson probability formula:

\( \textbackslash Pr(X=k) = \frac{\textbackslash e^{-\textbackslash\textlambda}\textbackslash\textlambda^k}{k!} \)

Plugging in the values for part (a):

\( \textbackslash Pr(X=7)= \frac{\textbackslash e^{-7}7^7}{7!} \)

Calculate this to get approximately 0.149.

This means there's about a 14.9% chance of exactly seven hits occurring between 7:30 p.m. and 7:35 p.m.

Understanding the expected value helps in predicting the average outcome over many trials. In a Poisson distribution, the expected value (mean) is directly related to the rate of occurrence and the time period.

Here, hits occur at a rate of 1.4 per minute.

The expected number of hits (\textbackslash\textlambda) for 5 minutes is calculated as:

\( \textbackslash\textlambda = 1.4 \textbackslash times 5 = 7 \)

This means that on average, you would expect 7 hits every 5 minutes.

The expected value tells us what to anticipate over a long period or numerous trials, even if individual occurrences vary.

In this example, despite random variations, the long-term average number of hits in each 5-minute interval would be around 7.

Moving on to cumulative probabilities, which help us understand the chance of observing up to a certain number of events.

For part (b), we need to calculate the probability of fewer than seven hits. This is done by summing individual probabilities from zero to six:

\( \textbackslash Pr(X < 7) = \textbackslash sum_{k=0}^{6} \textbackslash Pr(X=k) \)

While it's time-consuming to do manually, we can use tables or calculators for cumulative Poisson probabilities.

The computation gives us approximately 0.503.

This means there's about a 50.3% chance of having fewer than seven hits between 7:30 p.m. and 7:35 p.m.

For part (c), we need to find the probability of at least seven hits.

Using the complement rule, we get:

\( \textbackslash Pr(X \textbackslash ge 7) = 1 - \textbackslash Pr(X < 7) = 1 - 0.503 = 0.497 \)

This indicates a 49.7% chance of having at least seven hits in the same time period.