Skill: Determine theoretical probability

Theoretical probability tells how likely an event is when all outcomes are equally likely.

Use the formula

$$P(E)=\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}.$$

Example

A fair six-sided die is rolled. What is the probability of rolling an even number?

Step 1: List all possible outcomes.

A die can show

$$1,2,3,4,5,6$$

There are $6$ total outcomes.

Step 2: Identify the favorable outcomes.

The even numbers are

$$2,4,6$$

There are $3$ favorable outcomes.

Step 3: Use the probability formula.

$$P(\text{even})=\frac{3}{6}$$

Step 4: Simplify and represent the answer in other forms.

$$\frac{3}{6}=\frac{1}{2}=0.5=50\%$$

So, the theoretical probability of rolling an even number is

$$\boxed{\frac{1}{2},\text{ or }0.5,\text{ or }50\%}.$$

This is a theoretical probability because it is based on the possible outcomes of a fair die. The actual results from rolling a die many times may not be exactly $(50\%)$.