This problem deals with binomial distribution because each patient call is an independent event with only two possible outcomes: the patient either has the flu or doesn't have the flu. We can define the situation using the following parameters:

• $X$ is the random variable representing the number of patients (out of 30) that actually have the flu.
• Each patient has a 16% chance of having the flu, so the probability of success ($p$) is 0.16.
• There are $n = 30$ patients who call, and we want to find probabilities for specific values of $X$.

The distribution of $X$ can be described as:

$X \sim \text{Binomial}(n = 30, p = 0.16)$

1. Finding the probability that exactly 4 of the 30 patients actually have the flu:

We use the binomial probability formula to find this probability. The formula is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

Where:

• $n = 30$,
• $k = 4$ (since we are looking for the probability that exactly 4 patients have the flu),
• $p = 0.16$, and
• $1 - p = 0.84$.

Let's calculate $P(X = 4)$.

2. Finding the probability that fewer than 8 patients actually have the flu:

Here, we need to calculate $P(X < 8)$, which is the cumulative probability of $X$ taking values from 0 to 7. This can be calculated as:

$P(X < 8) = P(X = 0) + P(X = 1) + \dots + P(X = 7)$

We can compute this cumulative probability directly. I'll calculate both now.1. The probability that exactly 4 of the 30 patients actually have the flu is approximately 0.193 (or 19.3%).

2. The probability that fewer than 8 of the 30 patients actually have the flu is approximately 0.905 (or 90.5%).

Would you like more details on how these calculations are done, or do you have any other questions?

Here are 5 related questions that expand on this information:

1. What is the probability that none of the 30 patients actually have the flu?
2. What is the expected number of patients who actually have the flu out of 30?
3. How would the probabilities change if the probability of having the flu increased to 25%?
4. What is the probability that more than 10 of the 30 patients actually have the flu?
5. Can we approximate this binomial distribution with a normal distribution? If so, how?

Tip: For large sample sizes, binomial distributions can often be approximated by a normal distribution, which simplifies calculations.