To find the probability that at least 7 out of 10 customers will order non-alcoholic beverages when the probability of ordering non-alcoholic beverages is 0.51

To solve this problem, we need to calculate the probability that at least 7 out of 10 customers will order non-alcoholic beverages, given that the probability of a customer ordering a non-alcoholic beverage is 0.51. This scenario can be modeled using the binomial distribution.

### Binomial Distribution Formula

The probability of getting exactly \( k \) successes in \( n \) independent Bernoulli trials is given by the formula:

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

Where:
- \( n \) = number of trials (in this case, 10 customers)
- \( k \) = number of successes (number of customers ordering non-alcoholic beverages)
- \( p \) = probability of success on a single trial (0.51)
- \( \binom{n}{k} \) = binomial coefficient

### Required Probability
We need to find the probability that at least 7 customers order non-alcoholic beverages:

\[
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

### Step-by-Step Calculation

1. **Calculate \( P(X = 7) \)**:

\[
P(X = 7) = \binom{10}{7} \cdot 0.51^7 \cdot (1-0.51)^{3}
\]

2. **Calculate \( P(X = 8) \)**:

\[
P(X = 8) = \binom{10}{8} \cdot 0.51^8 \cdot (1-0.51)^{2}
\]

3. **Calculate \( P(X = 9) \)**:

\[
P(X = 9) = \binom{10}{9} \cdot 0.51^9 \cdot (1-0.51)^{1}
\]

4. **Calculate \( P(X = 10) \)**:

\[
P(X = 10) = \binom{10}{10} \cdot 0.51^{10} \cdot (1-0.51)^{0}
\]

5. **Sum the Probabilities**:
Add up the probabilities for \( P(X = 7) \), \( P(X = 8) \), \( P(X = 9) \), and \( P(X = 10) \).

Let's compute these values.The probability that at least 7 out of 10 customers will order non-alcoholic beverages, given that the probability of ordering non-alcoholic beverages is 0.51, is approximately **0.189** (or 18.88%).

Would you like more details or have any questions about this calculation?

Here are some related questions you might find interesting:

1. What is the probability that exactly 7 customers order non-alcoholic beverages?
2. How does the probability change if the probability of ordering non-alcoholic beverages increases to 0.6?
3. What is the expected number of customers ordering non-alcoholic beverages out of 10?
4. How would the probability change if we increased the number of customers to 20?
5. What is the variance of the number of customers ordering non-alcoholic beverages?
6. Can you explain the concept of the binomial coefficient used in the calculation?
7. How does this problem relate to the Central Limit Theorem?
8. What is the probability that fewer than 3 customers will order non-alcoholic beverages?

**Tip:** The binomial distribution is particularly useful for modeling situations where you have a fixed number of independent trials, each with the same probability of success.

Calculate the probability that at least 7 out of 10 customers will order non-alcoholic beverages, given a probability of 0.51 for ordering non-alcoholic beverages. Use the binomial distribution formula to solve this problem step-by-step.

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