Question

Records at the UH library show that 12% of all UH students check out books on history, 28% of all UH students check out books on science, and 6% check out books on both history and science. What is the probability that a randomly selected UH student checks out a history book or a science book or both?

Answer 1

There is a 34% probability that a randomly selected UH student checks out a history book or a science book or both.

Explanation:

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that a UH student checks out books on history.

B is the probability that a UH students checks out books on science.

We have that:

`A = a + (A \cap B)`

In which a is the probability that a UH student checks a book on history but not on science and
`A \cap B` is the probability that a UH student checks books both on history and science.

By the same logic, we have that:

`B = b + (A \cap B)`

What is the probability that a randomly selected UH student checks out a history book or a science book or both?

`P = a + b + (A \cap B)`

We start finding these values from the intersection.

6% check out books on both history and science. So
`A \cap B = 0.06`

28% of all UH students check out books on science. So
`B = 0.28`

`B = b + (A \cap B)`

`0.28 = b + 0.06`

`b = 0.22`

12% of all UH students check out books on history

`A = a + (A \cap B)`

`0.12 = a + 0.06`

`a = 0.06`

So

`P = a + b + (A \cap B) = 0.06 + 0.22 + 0.06 = 0.34`

There is a 34% probability that a randomly selected UH student checks out a history book or a science book or both.