Explain the Binomial Distribution with example.
The probability that student knows the correct answer to a multiple choice question is 2/3. If the student does not know the answer, the student guesses the answer. The probability of guessed answer is correct is ¼ . Given that student has answered the question correctly. Calculate the conditional probability that student knows the correct answer.

### Binomial Distribution Explanation

The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. A Bernoulli trial is an experiment where there are only two possible outcomes: success or failure.

The probability mass function (PMF) of the Binomial distribution is given by:

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

where:
- \( X \) is the random variable representing the number of successes,
- \( n \) is the number of trials,
- \( k \) is the number of successes,
- \( p \) is the probability of success on each trial,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

### Example Scenario: Multiple Choice Question

Let's address the problem step by step.

#### Given:
- Probability that the student knows the answer (\(P(K)\)) = \(\frac{2}{3}\)
- Probability that the student guesses (\(P(G)\)) = \(1 - P(K) = \frac{1}{3}\)
- If the student knows the answer, the probability of a correct answer (\(P(C \mid K)\)) = 1
- If the student guesses, the probability of a correct answer (\(P(C \mid G)\)) = \(\frac{1}{4}\)
- The student has answered the question correctly (\(C\)).

We need to find the conditional probability that the student knew the answer given that the student answered correctly, \(P(K \mid C)\).

### Bayes' Theorem

To solve this, we'll use **Bayes' Theorem**, which is stated as:

\[
P(K \mid C) = \frac{P(C \mid K) \cdot P(K)}{P(C)}
\]

Where \(P(C)\) is the total probability of answering correctly. Using the law of total probability:

\[
P(C) = P(C \mid K) \cdot P(K) + P(C \mid G) \cdot P(G)
\]

Substitute the given values:

\[
P(C) = 1 \cdot \frac{2}{3} + \frac{1}{4} \cdot \frac{1}{3}
\]

\[
P(C) = \frac{2}{3} + \frac{1}{12} = \frac{8}{12} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4}
\]

Now, substitute back into Bayes' Theorem:

\[
P(K \mid C) = \frac{1 \cdot \frac{2}{3}}{\frac{3}{4}} = \frac{\frac{2}{3}}{\frac{3}{4}} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}
\]

### Conclusion

The conditional probability that the student knew the correct answer given that they answered correctly is \(\frac{8}{9}\).

### Further Questions and Tip

1. How is the Binomial Distribution used in real-world scenarios?
2. What are the conditions required for a Binomial Distribution to be applicable?
3. Can you explain how to calculate the mean and variance of a Binomial Distribution?
4. How does the Binomial Distribution relate to the Normal Distribution?
5. What is the significance of Bayes' Theorem in probability theory?

**Tip:** When solving problems involving conditional probabilities, always check if Bayes' Theorem applies—it’s a powerful tool for reversing conditional dependencies.

The probability that a student knows the correct answer to a multiple-choice question is 2/3. If the student does not know the answer, the student guesses. The probability of a guessed answer being correct is 1/4. Given that the student has answered the question correctly, calculate the conditional probability that the student knows the correct answer.

Explore how to calculate the conditional probability that a student knows the correct answer to a multiple-choice question using Bayes' Theorem. This example demonstrates key concepts in probability, including conditional probability and the law of total probability. Suitable for students in Grades 11-12.

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