Question

A student received an A on the first test of the semester. The student wants to calculate the probability of scoring an A on the second test. Historically, the instructor knows that the joint probability of scoring A's on the first two tests is 0.5. Also, historically, the probability that a student scores and "A" on the second test given that a student scored an A on the first test is 0.9. What is the probability that a student will score an A on the second test?

a) 0.50
b) 0.95
c) 0.55
d) 0.90

Answer 1

To calculate the probability of a student scoring an A on the second test based on the given historical probabilities, we use the formula for conditional probability, yielding an approximate probability of 0.55, which is option c).

Step-by-step explanation:

The question presented involves calculating the probability a student will score an A on the second test, using given historical data as a reference point. To approach this problem, we can utilize the concept of conditional probability.

According to the data provided, the probability of scoring an A on the first and second tests is 0.5 (P(A1 and A2) = 0.5). Additionally, given that an A was scored on the first test, the probability of scoring an A on the second test is 0.9 (P(A2|A1) = 0.9). To calculate the unconditional probability of scoring an A on the second test (P(A2)), we can use the formula for conditional probability: P(A2|A1) = P(A1 and A2) / P(A1).

We are given P(A2|A1) and P(A1 and A2), but we need to find P(A1) to solve for P(A2). Rearranging the formula gives us P(A1) = P(A1 and A2) / P(A2|A1).

Since P(A1 and A2) = 0.5 and P(A2|A1) = 0.9, we find that P(A1) = 0.5 / 0.9 ≈ 0.55 (or 55%).

Hence, the probability that a student will score an A on the second test is approximately 0.55, which corresponds to option c).