Of all the videotapes in a large shipment, 21% have a defective tape, 16% have a defective case, and 11% have both defects. If you purchase one of the videotapes in this shipment, find the probability that it has the following.

Question

Of all the videotapes in a large shipment, 21% have a defective tape, 16% have a defective case, and 11% have both defects. If you purchase one of the videotapes in this shipment, find the probability that it has the following.

(a) A defective tape or a defective case.
p = .26

(b) A good tape or a good case.
p = ?

(c) A good tape and a good case.
p = .74

Answer 1

Out of 100

10 have only broken tape
5 have only broken case
11 have both broken
so
26 have something broken
74 are whole
100 - 11 = 89 have a good case or a good tape (or both)

Answer 2

To solve this problem, we need to use the concept of probability and the principle of inclusion-exclusion.

Let's break down the information given in the problem:

- 21% of the videotapes have a defective tape.
- 16% of the videotapes have a defective case.
- 11% of the videotapes have both defects.

(a) To find the probability that a tape has a defective tape or a defective case, we can simply add the probabilities of each defect and subtract the probability of having both defects (to avoid double counting):

P(defective tape or defective case) = P(defective tape) + P(defective case) - P(both defects)
= 21% + 16% - 11%
= 26%

So, the probability that a tape has a defective tape or a defective case is 26%.

(b) To find the probability that a tape has a good tape or a good case, we can use the complement rule. The complement of having a defective tape or a defective case is having a good tape and a good case:

P(good tape or good case) = 1 - P(defective tape or defective case)
= 1 - 26%
= 74%

So, the probability that a tape has a good tape or a good case is 74%.

(c) To find the probability that a tape has a good tape and a good case, we can subtract the probability of having a defective tape or a defective case from 100%:

P(good tape and good case) = 1 - P(defective tape or defective case)
= 1 - 26%
= 74%

So, the probability that a tape has a good tape and a good case is also 74%.

Therefore, we have:
(a) P(defective tape or defective case) = 26%
(b) P(good tape or good case) = 74%
(c) P(good tape and good case) = 74%