Binomial Probability in Bar Examination

Understanding the Problem

The problem describes a binomial experiment because it meets the following criteria:

• There are a fixed number of trials (n = 9 questions).
• Each trial is independent (the answer to one question doesn't affect others).
• There are only two possible outcomes for each trial (correct or incorrect).
• The probability of success (correct answer) is constant for each trial (p = 1/4 = 0.25).

Binomial Probability Formula

The probability of exactly k successes in n trials is given by the binomial probability formula:

P(X = k) = ⁿC_k * p^k * (1 - p)^{(n - k)}

Where:

• P(X = k) is the probability of exactly k successes
• ⁿC_k is the number of combinations of n items taken k at a time (also written as n! / (k! * (n-k)!))
• p is the probability of success on a single trial
• (1 - p) is the probability of failure on a single trial

Applying the Formula to the Question

In this case:

• n = 9 (number of questions)
• k = 6 (number of correct answers)
• p = 0.25 (probability of a correct answer)
• (1 - p) = 0.75 (probability of an incorrect answer)

Calculating ⁹C₆

⁹C₆ = 9! / (6! * 3!) = (9 * 8 * 7) / (3 * 2 * 1) = 84

Calculating p^k and (1 - p)^{(n - k)}

p^k = (0.25)⁶ = 0.000244140625

(1 - p)^{(n - k)} = (0.75)³ = 0.421875

Calculating P(X = 6)

P(X = 6) = 84 * 0.000244140625 * 0.421875 = 0.0086498046875

Final Answer

Therefore, the probability that the student chooses exactly six correct answers is approximately 0.00865 or 0.865%.