Counting outcomes
Count every possible result with a table or a tree: two dice, totals like 7, and the orders of books on a shelf.
How many different codes does a bike lock have? How many outfits can you make from the clothes in your closet? In how many orders can five people stand in a line? These are counting questions, and the numbers grow surprisingly fast. Counting all the possible results is also the first step of many probability questions, because the total number of results is the bottom of the fraction. You do not need to write every result down. A few simple rules do the counting for you.
The multiplication rule
You have 3 shirts and 5 pairs of pants. Each shirt can go with each pair of pants. So the first shirt gives 5 outfits, the second shirt gives 5 more, and the third shirt 5 more: 5 + 5 + 5 = 3 × 5 = 15 outfits.
Remember
If one choice can be made in a ways and a second choice in b ways, the two together can be made in a × b ways.
For three choices, multiply three numbers, and so on.
A tree diagram shows the same idea as a drawing. For a coin and then a die, draw 2 branches for the coin (H and T). From the end of each one, draw 6 branches for the die. At the end there are 2 × 6 = 12 branch tips, one for each result: H1, H2, ..., H6, T1, T2, ..., T6.
| What you do | Working | Number of results |
|---|---|---|
| Flip a coin and roll a die | 2 × 6 | 12 |
| Roll two dice | 6 × 6 | 36 |
| Flip a coin 3 times | 2 × 2 × 2 | 8 |
| Flip a coin 4 times | 2 × 2 × 2 × 2 | 16 |
| Roll a die 3 times | 6 × 6 × 6 | 216 |
All 36 results of two dice
Roll a red die and a blue die and add the numbers. The table shows all 36 results. The red die is down the side and the blue die is across the top. Each cell is the total.
| Red down, blue across | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Count how many cells show each total. The totals go from 2 to 12, but they are NOT equally likely. The middle total, 7, appears 6 times. The ends, 2 and 12, appear only once each.
| Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
So the probability of a total is the number of ways divided by 36. A total of 7 has probability 6/36 = 1/6, and 7 is the most likely total of all.
Example 1
You roll two dice and add the numbers. What is the probability that the total is 6?
- List the ways: 1+5, 2+4, 3+3, 4+2, 5+1.
- That is 5 of the 36 results.
- 5 and 36 have no common factor, so the fraction is already simple.
Answer: 5/36
Example 2
You roll two dice and add the numbers. What is the probability that the total is 4 or less?
- Total 2: 1 way (1+1).
- Total 3: 2 ways (1+2, 2+1).
- Total 4: 3 ways (1+3, 2+2, 3+1).
- 1 + 2 + 3 = 6 ways, so 6/36 = 1/6.
Answer: 1/6
Putting things in order
In how many orders can you put 4 different books on a shelf? Fill the places one at a time. Any of the 4 books can go first. Then 3 books are left for the second place, 2 for the third, and 1 for the last.
Remember
Orders of n different things: n × (n − 1) × ... × 2 × 1
This product has a name: n factorial, written n!. So 4! = 4 × 3 × 2 × 1 = 24.
Example 3
In how many different orders can you write the letters D, O and G?
- 3 choices for the first letter, 2 for the second, 1 for the last.
- 3 × 2 × 1 = 6.
- Check by listing: DOG, DGO, ODG, OGD, GDO, GOD.
Answer: 6
The numbers grow fast. 5 people can stand in a line in 5 × 4 × 3 × 2 × 1 = 120 ways, and 10 people in more than 3.6 million ways.
Sometimes you only fill some of the places. In a race with 5 runners, how many ways can the first and second places be filled? Any of the 5 can be first, then any of the other 4 can be second: 5 × 4 = 20. You stop after two places.
When order does not matter
You choose 2 friends out of 5 to come to the cinema. If you count as before, you get 5 × 4 = 20. But choosing Ana and then Ben gives the same pair as Ben and then Ana. Every pair has been counted twice, so divide by 2: 20 ÷ 2 = 10 pairs.
Example 4
7 people meet. Each person shakes hands once with every other person. How many handshakes are there?
- Each of the 7 people shakes 6 hands: 7 × 6 = 42.
- But each handshake has two people in it, so it was counted twice.
- 42 ÷ 2 = 21.
Answer: 21 handshakes
Codes where digits can repeat
A lock has 3 wheels, and each wheel shows a digit from 0 to 9. A digit can be used again, so each wheel has 10 choices: 10 × 10 × 10 = 1000 codes. They are the numbers 000 to 999, which is 1000 numbers, not 999.
Try it yourself
1.A cafe offers 4 drinks and 3 cakes. How many different ways are there to choose one drink and one cake?
Show the answer
12
Each of the 4 drinks goes with each of the 3 cakes: 4 × 3 = 12.
2.You roll two dice and add the numbers. What is the probability that the total is 5?
Show the answer
1/9
1+4, 2+3, 3+2 and 4+1 work. That is 4 of the 36 results: 4/36 = 1/9.
3.In how many different orders can you write the letters of the word MATH?
Show the answer
24
4 different letters: 4 × 3 × 2 × 1 = 24.
4.In a race with 6 runners, how many ways can the gold, silver and bronze medals be given?
Show the answer
120
6 choices for gold, then 5 for silver, then 4 for bronze: 6 × 5 × 4 = 120. The order matters here.
5.You choose 2 colors out of 7 to paint a room. How many different pairs can you choose?
Show the answer
21
7 × 6 = 42 counts each pair twice, so 42 ÷ 2 = 21.
6.A phone code has 4 digits, each from 0 to 9. How many codes are there?
Show the answer
10000
10 choices for each digit: 10 × 10 × 10 × 10 = 10000.
Ready to test yourself?
Play Probability: 20 seconds per question, and it gets harder as you go.