The probability that a student passes a physics test is 23 2 3 . If he takes three physics tests, what is the probability that he passes two of the tests?

The probability that a student passes a physics test is $\frac{2}{3}$. If he takes three physics tests, what is the probability that he passes two of the tests?

Answer Details

The probability that the student passes a physics test is 2/3. Let's call this probability p. The probability that he fails a physics test is therefore 1 - p = 1/3. To find the probability that he passes two out of three tests, we need to consider the different ways in which this can happen. He could pass the first two tests and fail the third, or pass the first and third tests but fail the second, or pass the second and third tests but fail the first. Each of these scenarios has the same probability: p * p * (1 - p) + p * (1 - p) * p + (1 - p) * p * p Simplifying this expression, we get: 3p^2(1-p) Substituting p = 2/3, we get: 3(2/3)^2(1-2/3) = 4/27 Therefore, the probability that the student passes two out of three physics tests is 4/27, which corresponds to option (C).