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The problem in part (ii) asks us to find the probability that **less than 8 individuals ask for water with their meal** from a random sample of 10 individuals. 

We know that:
- On average, \( \frac{2}{5} \) of individuals ask for water, which means the probability of an individual asking for water is \( p = 0.4 \).
- We are working with a sample of 10 individuals, so \( n = 10 \).

This is a **binomial probability problem** because there are two possible outcomes (either someone asks for water or they don't), and the probability remains constant for each individual.

### Binomial Distribution
The binomial probability formula is:

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

where:
- \( P(X = k) \) is the probability of exactly \( k \) successes (in this case, people asking for water),
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \),
- \( p \) is the probability of a success (asking for water, \( 0.4 \) in this case),
- \( n \) is the total number of trials (10 individuals),
- \( k \) is the number of successes we're interested in.

For part (ii), we are asked to find the probability that **less than 8 individuals ask for water**, i.e., \( P(X < 8) \).

### Step-by-Step Calculation
This means we need to calculate the cumulative probability:

\[
P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + \dots + P(X = 7)
\]

Instead of calculating each term manually, we can utilize the binomial cumulative distribution function (CDF). The CDF gives us the sum of probabilities up to a certain number of successes.

Thus, we need to compute \( P(X \leq 7) \), which is the cumulative probability for \( X = 7 \).

### Using a calculator or software, we compute:

\[
P(X \leq 7) \approx 0.967
\]

Thus, the probability that fewer than 8 people ask for water with their meal is approximately **0.967** or **96.7%**.

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Would you like further details or clarification on any steps?

Here are some related questions to deepen your understanding:
1. How would the probability change if the sample size were 15 instead of 10?
2. What would the probability be if exactly 7 people asked for water?
3. Can we approximate this binomial distribution with a normal distribution?
4. How would the solution differ if the probability of asking for water was \(0.6\)?
5. How can the binomial theorem be used to calculate cumulative probabilities?

**Tip**: For binomial problems with large \( n \), using cumulative distribution tables or software tools makes calculations quicker and more accurate.

In a restaurant, an average of 2 out of every 5 individuals ask for water with their meal. A random sample of 10 individuals is selected. Find the probability that (ii) less than 8 ask for water with their meal.

Solve a binomial probability problem where 2 out of 5 individuals ask for water. Find the probability that less than 8 individuals from a sample of 10 ask for water. This solution includes step-by-step calculations using the binomial distribution formula.

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