A club of 8 picks a president and a vice president at random, P(Ann…

Step by step

The question

Question: A club of 8 people picks a president and a vice president at random.

What is the probability that Ann is president?

The probability is the number of ways we want, over all the ways.

P stands for probability.

1 Order: yes or no

Does order matter?

Yes.

Ann president and Ben vice is not Ben president and Ann vice.

So we count permutations.

2 Count all the ways

8 P 2: 8 choices for the first place, then 7 for the second.

8 times 7 is 56 ways.

3 Count the ways we want

Ann president: 1 way.

Then any of the other 7 people can be vice.

1 times 7 is 7.

4 Write the fraction

7 ways we want, out of 56.

Careful!

Not 8 C 2: that counts groups.

Here the places are different, so order matters.

5 Simplify

Both 7 and 56 can be divided by 7.

That makes it simpler.

So it is 1/8.

6 Check

Let's check another way: one place at a time.

Ann president: 1 out of 8.

Then anyone left can be vice: 7 out of 7.

Multiplied, that is 1/8.

The same.

So the answer is right!

Let's look back.

First: order: yes or no.

Then: count all the ways.

Then: count the ways we want.

Then: write the fraction.

Then: simplify.

Last: check.