Calculating probabilities
In a nutshell
Probabilities describe the likelihood of an event occurring. They are quantities that are always between 0 and 1.
Definitions
Here are some definitions that you need to know.
term | definition |
Experiment | A repeatable process. |
Event | One or more possible outcomes of an experiment. |
Sample space | The set of all possible outcomes of an experiment. |
Representing sample space
The sample space of an experiment can be represented with a sample space diagram. These are an easy way to list possible outcomes.
Example 1
A fair coin is tossed 3 times. What is the sample space of this experiment?
Let H represent heads, and T represent tails.
The set of possible options is:
{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
Example 2
Two fair dice are tossed, and the numbers that they land on are multiplied together. What is the probability that the two numbers multiply to give 12?
Draw a table to represent the sample space:
| die 1 |
die 2 | | 1 | 2 | 3 | 4 | 5 | 6 |
1 | 1 | 2 | 3 | 4 | 5 | 6 |
2 | 2 | 4 | 6 | 8 | 10 | 12 |
3 | 3 | 6 | 9 | 12 | 15 | 18 |
4 | 4 | 8 | 12 | 16 | 20 | 24 |
5 | 5 | 10 | 15 | 20 | 25 | 30 |
6 | 6 | 12 | 18 | 24 | 30 | 36 |
Identify the desired events in the sample space:
| die 1 |
die 2 | | 1 | 2 | 3 | 4 | 5 | 6 |
1 | 1 | 2 | 3 | 4 | 5 | 6 |
2 | 2 | 4 | 6 | 8 | 10 | 12◯ |
3 | 3 | 6 | 9 | 12◯ | 15 | 18 |
4 | 4 | 8 | 12◯ | 16 | 20 | 24 |
5 | 5 | 10 | 15 | 20 | 25 | 30 |
6 | 6 | 12◯ | 18 | 24 | 30 | 36 |
Calculate the probability:
4 outcomes out of 6×6=36 options.
The probability that the two numbers multiply to give 12 is 364=91.