This problem involves a binomial distribution, as it deals with repeated independent trials (each customer deciding to buy or not buy a magazine) with two possible outcomes: success (buying a magazine) or failure (not buying a magazine).

In a binomial distribution, the probability of exactly $k$ successes in $n$ independent trials is given by the formula:

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

Where:

• $P(X = k)$ is the probability of getting exactly $k$ successes,
• $\binom{n}{k}$ is the binomial coefficient, which counts the number of ways to choose $k$ successes from $n$ trials,
• $p$ is the probability of success on each trial,
• $n$ is the total number of trials, and
• $k$ is the number of successes.

In this case:

• $n = 10$ (since he is observing the first 10 customers),
• $k = 3$ (the problem asks for the probability that exactly 3 customers buy a magazine),
• $p = 0.079$ (since 7.9% of customers buy a magazine).

We can now calculate the probability.

Step 1: Calculate the binomial coefficient $\binom{10}{3}$:

$\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

Step 2: Apply the binomial formula:

$P(X = 3) = \binom{10}{3} (0.079)^3 (1 - 0.079)^{10-3}$ $P(X = 3) = 120 \times (0.079)^3 \times (0.921)^7$

Let me calculate this.The probability that exactly 3 out of the first 10 customers will buy a magazine is approximately $0.0333$, or about 3.33%.

Would you like to see more details or have any questions about this process?

Here are 5 related questions to further explore the topic:

1. What is the probability that no customers buy a magazine?
2. How would the probability change if the deli owner suspects that 10% of customers buy a magazine?
3. What is the probability that fewer than 2 customers buy a magazine?
4. How can we calculate the expected number of customers who buy a magazine in this scenario?
5. What is the standard deviation of the number of customers buying a magazine?

Tip: The sum of probabilities for all possible outcomes (0 through 10 customers buying a magazine) must always equal 1 for a binomial distribution.