Question

A student has a box containing 25 computer disks, of which 15 are blank and 10 are not. She randomly selects disks one by one and examines each one, terminating the process only when she finds a blank disk. What is the probability that she must examine at least two disks? (Hint: What must be true of the first disk?)

Short answer

The probability that the student must examine at least two disks to find a blank one is 0.25 or 25%.

Step-by-step solution

Identify the probabilities

Firstly, the student chooses a disk from the box which contains 25 disks, out of which 10 are not blank. Hence, the probability to pick a non-blank disk first is given by the ratio of the number of non-blank disks to the total number of disks. Thus, \( P_1 = \frac{10}{25} = 0.4 \).

Conditional Probability

Secondly, as one disk has already been picked and it was not blank, the conditions change for the next selection. Now there are 24 disks remained, out of which 15 are blank. Therefore, the student's successful attempt in the next picking, or finding a blank disk on the second pick, is given by \( P_2 = \frac{15}{24} = 0.625 \).

Compute the final probability

The two events described are independent; the result of the first pick does not affect the second one. As the student must examine at least two disks, this means that the first disk she picks up needs to be a non-blank one and the second one should be blank. Therefore, final probability will be the product of \( P_1 \) and \( P_2 \). Consequently, the final probability \( P = P_1 * P_2 = 0.4 * 0.625 = 0.25 \).