A teacher selects a student at random from his class of 20 students every week for 20 consecutive weeks. What is the probability that a particular student, Charlie, is selected exactly three times?

This problem involves calculating the probability of a specific outcome in a binomial experiment. Since Charlie either is selected or is not selected each week, this is a binomial probability scenario.

### Binomial Probability Formula

The binomial probability formula is:

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

where:
- \(n\) is the number of trials (weeks),
- \(k\) is the number of successes (times Charlie is selected),
- \(p\) is the probability of success in each trial,
- \(\binom{n}{k}\) is the binomial coefficient (combination), which is calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

### Given Information
- The number of trials \(n = 20\) (since there are 20 weeks),
- The probability of success (Charlie being selected in a given week) \(p = \frac{1}{20}\),
- The number of successes \(k = 3\) (Charlie is selected exactly 3 times).

### Step-by-Step Calculation

1. **Calculate the binomial coefficient:**

\[
\binom{20}{3} = \frac{20!}{3!(20 - 3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]

2. **Calculate the probability for \(k = 3\):**

The probability that Charlie is selected in any one week is \(p = \frac{1}{20}\), and the probability that Charlie is not selected is \(1 - p = \frac{19}{20}\).

Thus, the binomial probability is:

\[
P(X = 3) = \binom{20}{3} \left(\frac{1}{20}\right)^3 \left(\frac{19}{20}\right)^{20 - 3}
\]

\[
P(X = 3) = 1140 \times \left(\frac{1}{20}\right)^3 \times \left(\frac{19}{20}\right)^{17}
\]

\[
P(X = 3) = 1140 \times \frac{1}{8000} \times \left(\frac{19}{20}\right)^{17}
\]

Now, let's compute the value of \(\left(\frac{19}{20}\right)^{17}\) and finalize the calculation.

I'll do this calculation for you.The probability that Charlie is selected exactly 3 times in 20 consecutive weeks is approximately \(0.0596\), or about 5.96%.

Would you like further details or explanations? Here are five related questions to consider:

1. How does the probability change if the number of weeks increases to 30?
2. What is the probability that Charlie is selected at least once in 20 weeks?
3. How would the probability change if Charlie were selected exactly 5 times instead of 3?
4. What is the expected number of times Charlie will be selected in 20 weeks?
5. How would you calculate the probability that Charlie is never selected?

**Tip**: Binomial distribution is a great tool for solving problems with fixed numbers of independent trials and a constant probability of success in each trial.

Discover how to calculate the binomial probability of Charlie being selected exactly 3 times out of 20 weeks. This problem covers the binomial distribution, probability calculation, and combination formula. Ideal for students in Grades 10-12 looking to understand real-world probability scenarios.

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