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# Probability and Statistics

# How to Solve a Math Problem

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

### Sample Problem

Lia has 15 books but only likes two thirds of them. If she picks two books at random to take on a plane ride, what is the probability that she will like both of the books she picks? Note to Lia: have a yard sale when you go back home.

1. Figure out what the problem is asking.

This problem is asking about "picking" or "choosing" books. Since nothing is said about order, the problem is asking about combinations rather than permutations. We're also being asked about probability, so we need to find a probability.

2. Solve the problem.

We want to find a probability. What is the probability that Lia likes both books she picks? The formula we remember for probability is

.

Let's look at the number of possible outcomes first. Lia is picking two books at random. The number of ways to do this, if we don't care about order, is

An outcome is "favorable" if Lia likes both the books she picks. She likes two thirds, or 10, of her books. If both the books she chooses come from these 10, then the outcome is favorable. The number of ways to choose her 2 books from the 10 she likes is

Putting it all together, the probability Lia likes both her books is

.

If she winds up with two books she *doesn't* like, her temperament on that plane will probably be a little...shall we say...turbulent.

3. Check the answer.

It can be a little tricky to check the answer for this sort of problem. When finding a probability, we can at least make sure our answer is between 0 and 1.

Since

,

we can at least be sure we've found a probability. It's all about the small victories.