8.02 Theoretical probability as a number

The likelihood of an event after a trial can be placed on a spectrum from 0 to 1 using fractions or decimals, or from 0\% to 100\% using percentages:

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 will now look at how to determine these numbers exactly.

An outcome represents a possible result of a trial. When you roll a six-sided die, the outcomes are the numbers from 1 to 6.

An event is a grouping of outcomes. When you roll a six-sided die, events might include "rolling an even number", or "rolling more than 5".

Each outcome is always an event - for example, "rolling a 5" is an event.

But other events might not match the outcomes at all, such as "rolling more than 6".

If every outcome in a trial is equally likely, then the probability of one particular outcome is given by the equation:\text{Probability} = \dfrac{1}{\text{Size of sample space}}

Remember that the sample space is the list of all possible outcomes. To find the probability as a percentage, convert the fraction to a decimal and then multiply by 100\%

Examples

If the outcomes in a sample space are not equally likely, then we have to think about splitting the sample space up into "favorable outcomes" and the rest. 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}}

If every outcome is favorable, then we have a probability of 1. If there are no favorable outcomes, then probability is 0.

Examples

Example 3

What is the probability of spinning a Star or an Apple on this spinner? Express your answer as a decimal.

Worked Solution

Create a strategy

Find the number of favorable outcomes and use the probability formula.

Apply the idea

There are 10 different sectors, 3 out of 10 are stars and 3 out of 10 are apples. There are 3 + 3 =6 favorable outcomes all together.

\displaystyle \text{Probability}	\displaystyle =	\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}	Use the formula
	\displaystyle =	\displaystyle \frac{6}{10}	Substitute the known values
	\displaystyle =	\displaystyle 0.6	Evaluate into decimal

So, probability of spinning a Star or an Apple is 0.6.

Example 4

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.

a

If the probability of picking a red marble is \dfrac{4}{10}, how many red marbles are there in the jar?

Worked Solution

Apply the idea

We know that the probability of picking a red marble is \dfrac{4}{10}. That means 4 out of 10 marbles are red. So there are 4 red marbles.

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b

What is the probability of picking a blue marble?

Worked Solution

Create a strategy

Find the number of blue marbles and write the probability as a fraction.

Apply the idea

There are 10 marbles in total. We know that 4 of them are red and the rest are blue.

So, there are 10 - 4 = 6 blue marbles.

So the probability is of picking a blue marble is \dfrac{6}{10}=\dfrac{3}{5}.

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