Combined events
Listing or counting all the possible outcomes for two or more combined events enables you to calculate the probability of any particular event occurring.
This can be done by listing outcomes systematically, using the product rule to calculate the number of outcomes, or using sample space diagrams to record all the outcomes in a table.
Systematic listing
The outcomes for an event can be listed in an organised or systematic way to make sure that none of the possible outcomes is missed out. Look for patterns to help find all the outcomes.
Example
How many different three-digit numbers can be made using the digits 3, 6, and 9?
First, find all the numbers that can start with 3:
- 369 and 396
There will also be two numbers that start with 6 and 9:
- 639 and 693
- 936 and 963
So there are six possible numbers:
- 369
- 396
- 639
- 693
- 936
- 963
The product rule for counting - Higher
To find the total number of outcomes for two or more events, multiply the number of outcomes for each event together. This is called the product rule for counting because it involves multiplying to find a product.
Example
A restaurant menu offers 4 starters, 7 main courses and 3 different desserts. How many different three-course meals can be selected from the menu?
Multiplying together the number of choices for each course gives \(4 \times 7 \times 3 = 84\) different three-course meals.
Sample space diagrams
Sample space diagrams are a visual way of recording the possible outcomes of two events, which can then be used to calculate The extent to which something is likely to be the case..
The tables include the possible outcomes of one event listed across and one event listed down.
Example
Two fair dice are rolled at the same time and their scores are added together. Find the probability of the To add. The sum of 8 and 4 is 12 as 8 + 4 = 12. of the two dice equalling 7.
Below is a table with the outcome of rolling die 1 across the top and die 2 down the left hand side.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1 + 1 = 2 | 1 + 2 = 3 | 1 + 3 = 4 | 1 + 4 = 5 | 1 + 5 = 6 | 1 + 6 = 7 |
| 2 | 2 + 1 = 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 |
| 1 | |
|---|---|
| 1 | 1 + 1 = 2 |
| 2 | 1 + 2 = 3 |
| 3 | 1 + 3 = 4 |
| 4 | 1 + 4 = 5 |
| 5 | 1 + 5 = 6 |
| 6 | 1 + 6 = 7 |
| 2 | |
|---|---|
| 1 | 2 + 1 = 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 6 |
| 5 | 7 |
| 6 | 8 |
| 3 | |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 4 | 7 |
| 5 | 8 |
| 6 | 9 |
| 4 | |
|---|---|
| 1 | 5 |
| 2 | 6 |
| 3 | 7 |
| 4 | 8 |
| 5 | 9 |
| 6 | 10 |
| 5 | |
|---|---|
| 1 | 6 |
| 2 | 7 |
| 3 | 8 |
| 4 | 9 |
| 5 | 10 |
| 6 | 11 |
| 6 | |
|---|---|
| 1 | 7 |
| 2 | 8 |
| 3 | 9 |
| 4 | 10 |
| 5 | 11 |
| 6 | 12 |
The sample space diagram shows there are 6 ways of making a 7, out of a total of 36 possible outcomes.
Therefore, the probability of rolling two dice and the sum being 7 is \(\frac{6}{36}\) = \(\frac{1}{6}\).
Frequency Trees
A frequency tree can be used to record and organise information given as frequencies. This can then be used to calculate probabilities.
Example
A running club has 160 members. 74 of the club members are female. 58 of the female members are over 18. 21 of the male club members are under 18.
Complete the frequency tree to show this information.
Find the probability that a member of the club chosen at random is under 18.
Since 74 members are female, \(160 - 74 = 86\) members must be male.
58 female members are over 18, so \(74 - 58 = 16\) females are under 18.
21 male members are under 18, so \(86 - 21 = 65\) males are over 18.
Adding this information to the frequency tree gives:
The total number of under 18s is \(16 + 21 = 37\), so the probability that a member of the club chosen at random is under 18 is \(\frac {37}{160}\).
