Multinomial Distribution: If a teacher randomly grades 10 papers and there’s a 40% chance of passing a paper, what is the probability of passing exactly 7 papers?

The scenario described can be modeled using the **Binomial distribution** rather than a Multinomial distribution. The reason is that the outcome of each paper is either "pass" or "fail," making it a binomial situation.

### Problem breakdown:
- **Number of trials** (\(n\)): 10 papers
- **Probability of success** (\(p\)): 40% or 0.4 (the chance of passing a paper)
- **Number of successes** (\(k\)): 7 papers need to pass
- **Probability of failure** (\(q = 1 - p\)): 60% or 0.6

The **Binomial probability formula** is:

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

where:
- \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\),
- \(p\) is the probability of success,
- \(k\) is the number of successful outcomes.

### Applying the formula:

\[
P(X = 7) = \binom{10}{7} (0.4)^7 (0.6)^{3}
\]

1. **Calculate the binomial coefficient**:

\[
\binom{10}{7} = \frac{10!}{7!3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

2. **Calculate the probability components**:
   - \(p^7 = (0.4)^7 = 0.0016384\),
   - \((1 - p)^{10 - 7} = (0.6)^3 = 0.216\).

3. **Calculate the final probability**:

\[
P(X = 7) = 120 \times 0.0016384 \times 0.216
\]

Let me calculate the final result for you.The probability of exactly 7 out of 10 papers passing, given a 40% chance of passing each paper, is approximately **0.0425** (or 4.25%).

Would you like more details, or do you have any further questions?

Here are 5 related questions you might find useful:
1. What is the probability of passing fewer than 7 papers?
2. How does the probability change if the chance of passing increases to 50%?
3. What is the expected number of passing papers in this situation?
4. Can we calculate the variance and standard deviation for the number of passing papers?
5. How would the probability change if the number of papers graded increased to 15?

**Tip:** The sum of all possible probabilities in a binomial distribution always equals 1.

Find out how to calculate the probability of passing exactly 7 out of 10 papers, given a 40% chance of passing each paper. This problem covers binomial distribution, probability concepts, and combinatorics. Suitable for high school students in Grades 9-12.

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