Skill: Conduct simple probability experiments

To conduct a simple probability experiment:

1. List all the possible outcomes.
2. Decide whether the outcomes are equally likely.
3. Describe the event you are studying.
4. Find the theoretical probability:

$$\text{Probability}=\frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}$$

5. Repeat the experiment and use the results to find the experimental probability:

$$\text{Experimental probability}=\frac{\text{number of times the event occurs}}{\text{number of trials}}$$

Example: Rolling a fair number cube

Suppose you want to find the probability of rolling an even number.

Step 1: List the possible outcomes.

A number cube can land on:

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

Step 2: Check whether the outcomes are equally likely.

Because the number cube is fair, each number has the same chance of being rolled.

Step 3: Identify the favorable outcomes.

The even numbers are:

$$2,4,6$$

There are $3$ favorable outcomes out of $6$ total outcomes.

Step 4: Find the theoretical probability.

$$P(\text{even})=\frac{3}{6}=\frac{1}{2}$$

So, the theoretical probability of rolling an even number is $\frac{1}{2}$.

Step 5: Try the experiment.

Roll the number cube $12$ times. Suppose the results are:

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

The even numbers occur $7$ times.

The experimental probability is:

$$\frac{7}{12}$$

The experimental probability, $\frac{7}{12}$, is close to the theoretical probability, $\frac{1}{2}$. They do not have to be exactly the same because a small number of rolls can have unusual results. If you roll the cube many more times, the experimental probability will generally get closer to $\frac{1}{2}$.