9 horses race, a bet picks the 1st and 2nd at random, P(the bet wins)

Step by step

The question

Question: 9 horses run a race.

A bet picks the 1st and 2nd places, in order, at random.

What is the probability that the bet wins?

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.

A first, B second is not B first, A second.

So we count permutations.

2 Count all the ways

9 P 2: 9 choices for the first place, then 8 for the second.

9 times 8 is 72 ways.

3 Count the ways we want

Only one order wins: the real 1st and 2nd.

So 1 way.

4 Write the fraction

1 way we want, out of 72.

Careful!

Not 9 C 2: that counts groups.

Here the places are different, so order matters.

5 Check

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

The right 1st: 1 out of 9.

Then the right 2nd: 1 out of the 8 left.

Multiplied, that is 1/72.

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.

Last: check.