megafan wrote:

A certain class consists of 8 students, including Kim. Each day, three tasks must be completed and are assigned as follows: one of the 8 students is selected at random to complete Task A, one of the remaining 7 students is selected at random to complete Task B, and one of the remaining six students is selected at random to complete Task C. What is the probability that Kim will be selected to complete one of the three tasks?

(A) \(\frac{1}{3}\)

(B) \(\frac{3}{8}\)

(C) \(\frac{1}{24}\)

(D) \(\frac{1}{336}\)

(E) \(\frac{1}{512}\)

The probability that Kim will be selected to complete one of the three tasks = P(Kim will be selected for task A) + P(Kim will be selected for task B) + P(Kim will be selected for task C)

Let’s calculate the probability that Kim will be selected for task A:

P(Kim will be selected for task A) = 1/8

Next, we calculate the probability that Kim will be selected for task B (i.e., she won’t be selected for task A):

P(Kim will be selected for task B) = P(Kim won’t be selected for task A) x P(Kim will be selected for task B) = 7/8 x 1/7 = 1/8

Finally, let’s calculate the probability that Kim will be selected for task C (i.e., she won’t be selected for either task A or task B):

P(Kim will be selected for task C) = P(Kim won’t be selected for task A) x P(Kim won’t be selected for task B) x P(Kim will be selected for task C) = 7/8 x 6/7 x 1/6 = 1/8

Thus, the probability that Kim will be selected for one of the three tasks is ⅛ + ⅛ + ⅛ = ⅜.

Alternate Solution:

We will use the fact that

P(Kim will be selected for one of the three tasks) + P(Kim will not be selected for any tasks) = 1

To calculate P(Kim will not be selected for any tasks), we note that this event happens if Kim is not selected for task A, B or C. Since there is a 1/8 probability that Kim is selected for task A, the probability that she is not selected for task A is 1 - 1/8 = 7/8. Similarly, the probability that she is not selected for tasks B and C are 6/7 and 5/6, respectively. Thus, the probability that she is not selected for any tasks is 7/8 x 6/7 x 5/6 = 5/8.

Now, since P(Kim will be selected for one of the three tasks) = 1 - P(Kim will not be selected for any tasks) and since P(Kim will not be selected for any tasks) = 5/8, we conclude that P(Kim will be selected for one of the three tasks) = 1 - 5/8 = 3/8.

Answer: B

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