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.