Theoretical probability
We can make predictions for trials by first creating the sample space and then determining the theoretical probability of each outcome.
If you roll two six-sided dice and add the numbers together, what is the probability of getting a sum of 6? What about a sum of 10 or greater? 8 or less?
Before we answer these questions we need to determine the sample space. The possible outcomes for two dice can be drawn in a grid:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1, 1 | 1, 2 | 1, 3 | 1, 4 | 1, 5 | 1, 6 |
| 2 | 2, 1 | 2, 2 | 2, 3 | 2, 4 | 2, 5 | 2, 6 |
| 3 | 3, 1 | 3, 2 | 3, 3 | 3, 4 | 3, 5 | 3, 6 |
| 4 | 4, 1 | 4, 2 | 4, 3 | 4, 4 | 4, 5 | 4, 6 |
| 5 | 5, 1 | 5, 2 | 5, 3 | 5, 4 | 5, 5 | 5, 6 |
| 6 | 6, 1 | 6, 2 | 6, 3 | 6, 4 | 6, 5 | 6, 6 |
We can now tell that there are 36 possible outcomes. Depending on the trial we can highlight the favorable outcomes corresponding to the event, and the probability of any particular event is given by the formula:\text{Probability} = \dfrac{\text{Number of favorable outcomes}}{36}
Exploration
Explore this applet to find the various probabilities:
Once we have a sample space with every outcome being equally likely, we can express the probability as a fraction, decimal, or percentage.
Examples
Example 1
A two-digit number is formed using the numbers 3 and 2. It can be two of the same or one of each number in any order.
What is the probability that the number formed is odd?
What is the probability that the number formed is more than 30?
To find the theoretical probability of an event use the formula: \text{Theoretical probability}=\dfrac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}
Repeating trials
Once we know the probability of an event, we can predict how many times this event will occur if a trial is repeated several times.
We multiply the probability of the event by the number of trials, rounding to the nearest whole number.
Examples
Example 2
An eight-sided die is rolled 25 times. How many times should we expect to roll a 7? Round your answer to the nearest whole number.
Example 3
A bag contains 28 red marbles, 27 blue marbles, and 26 black marbles.
What is the probability of drawing a blue marble?
A single trial is drawing a marble from the bag, writing down the colour, and putting it back. If this trial is repeated 400 times, how many blue marbles should you expect? Round your answer to the nearest whole number.
To predict how many times an event will occur if a trial is repeated several times, we multiply the probability of the event by the number of trials.