5.02 Theoretical probability
Theoretical probability
Probability is used to describe the likelihood of an event occurring. The probability of an event can be represented as a ratio or an equivalent fraction. It can be represented with a fraction or decimal between 0 and 1. It can also be represented by a percentage between 0\% and 100\%.
A probability can never be less than 0 or more than 1. The larger the number, the more likely it is, and the smaller the number, the less likely it is.
We can calculate the probability of an event by first creating the sample space and counting the number of possible outcomes. If the events in the sample space are equally likely, the ratio will be the same for each event.
For example, let's look at a full set of 52 playing cards:
There are 52 cards in the sample space, and each card has an equal chance of being drawn. The probablility of drawing any one card is \dfrac{1}{52}. Adding up the probabilities for each card \left(\dfrac{1}{52}+\dfrac{1}{52}+\dfrac{1}{52}+...+\dfrac{1}{52}=\dfrac{52}{52}\right) gives a sum of 1 or 100 \%.
If the outcomes in a sample space are not equally likely, then we have to find the number of favorable outcomes for the given event. Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. We can use the formula:\text{Theoretical probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
Let's look at events for the playing cards with outcomes that are not equally likely. For example, what is the probability of drawing a 7?
We already know that there are 52 possible outcomes, so we only need to determine the number of favorable outcomes. There are 4 cards with a 7 on them, so there are 4 favorable events. Therefore, the probability of drawing a 7 is \dfrac{4}{52}.
If every outcome is favorable, then it is certain to occur, so we will have a probability of 1. If there are no favorable outcomes, it is impossible for the event to occur, so the probability will be 0.
Here are some events sorted into each of the five likelihood categories:
| Event | Probability | Likehood |
|---|---|---|
| \text{Drawing a blue card} | 0:52 | \text{Impossible} |
| \text{Drawing a Spade} | \dfrac{13}{52} | \text{Unlikely} |
| \text{Drawing a black card } | 0.5 | \text{Equally likely} |
| \text{Drawing a number card }(2\text{ through }10) | 36:52 | \text{Likely} |
| \text{Drawing a card that is a Spade, Heart, Club, or Diamond } | 100\% | \text{Certain} |
Examples
Example 1
A bag contains 28 red marbles, 27 blue marbles, and 26 black marbles.
What is the probability of drawing a blue marble?
Example 2
The eight-sided die shown is rolled.
What is the chance of rolling a five or more?
What is the chance of rolling less than five? Write your answer as a percentage.
Example 3
What is the probability of spinning a Star or an Apple on this spinner? Express your answer as a decimal.
Example 4
A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.
If the probability of picking a red marble is \dfrac{4}{10}, how many red marbles are there in the jar?
What is the probability of picking a blue marble?
The theoretical probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.\text{Theoretical Probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
If every outcome is favorable, then we have a probability of 1. If there are no favorable outcomes, then the probability is 0.