Topic 6 of 8

Counting and probability

Step 13 · Algebra 2 topics

Count outcomes with the multiplication principle, permutations and combinations, and use counting to find probabilities, including for independent events and "at least one".

The counting principle

If one choice can be made in m ways and a second in n ways, the two together can be made in m × n ways.

Example: Outfits

4 shirts, 3 pairs of pants and 2 pairs of shoes. How many outfits are there?

4 × 3 × 2 = 24.

Answer24

Example: Codes

How many 4-digit PINs are there if digits can repeat?

10 choices for each of the 4 digits: 10 × 10 × 10 × 10.

Answer10,000

Example: No repeats

How many ways can 5 runners finish first, second and third?

5 choices for first, then 4, then 3.
5 × 4 × 3 = 60.

Answer60

Permutations and combinations

A permutation counts arrangements where order matters. A combination counts selections where order does not.

{}_{n}P_{r} = \frac{n!}{(n - r)!} \qquad {}_{n}C_{r} = \frac{n!}{r!\,(n - r)!}

Example: A permutation

How many ways can a club of 10 choose a president, a secretary and a treasurer?

Order matters: these are different jobs.
10 × 9 × 8 = 720.

Answer720

Example: A combination

How many ways can a club of 10 choose a committee of 3?

Order does not matter: each group of 3 was counted 3! = 6 times above.
720 ÷ 6 = 120.

Answer120

Probability with counting

When outcomes are equally likely, probability = favorable outcomes ÷ all outcomes. Counting methods find both numbers.

Example: A committee

A committee of 3 is chosen at random from 6 girls and 4 boys. What is the probability that it is all girls?

All girls: ${}_{6}C_{3} = 20$ ways.
All committees: ${}_{10}C_{3} = 120$ ways.
$\frac{20}{120}$ = $\frac{1}{6}$ .

Answer $\frac{1}{6}$

Example: Independent events

A coin is flipped and a number cube is rolled. What is the probability of heads and a 6?

For independent events, multiply: $\frac{1}{2} \times \frac{1}{6}$ .

Answer $\frac{1}{12}$

Example: At least one

What is the probability of at least one 6 in 2 rolls of a number cube?

The chance of no 6 is $\frac{5}{6} \times \frac{5}{6} = \frac{25}{36}$ .
So at least one 6 is 1 − $\frac{25}{36}$ .

Answer $\frac{11}{36}$

Try it yourself

Work each one on paper first, then open the answer to check.

A menu has 5 starters, 8 mains and 4 desserts. How many three-course meals are there?
Show answer
160
How many ways can 6 books be lined up on a shelf?
Show answer
720
How many ways can first and second prize go to 2 of 12 people?
Show answer
132
How many ways can 2 people be picked from 12?
Show answer
66
What is ${}_{7}C_{3}$ ?
Show answer
35
A bag has 3 red and 5 blue marbles. Two are drawn without putting the first back. What is the probability that both are red?
Show answer
$\frac{3}{28}$