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Empirical Probability

Identify favorable and total outcomes

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Theoretical Probability

Jessie has a spinner that is divided into four colors: blue, red, yellow and green. She knows that the spinner can be used to calculate theoretical probability, but she can’t remember what theoretical probability is. She is wondering how it is different from experimental probability. Can you help her by giving her an answer?

In this concept, you will learn how to define and calculate theoretical probability.

Theoretical Probability

Probability is defined as a mathematical way of calculating how likely an event is to occur. The probability of an event occurring is defined as the ratio of favorable outcomes to the number of possible equally likely total outcomes in a given situation.

In ratio form, the probability of an event is:

\begin{align*}P(\text{event})= \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes}\end{align*}

Theoretical probability is probability that is based on an ideal situation.

For instance, since a flipped coin has two sides and each side is equally likely to land up, the theoretical probability of landing heads (or tails) is exactly 1 out of 2. Whether or not the coin actually lands on heads (or tails) 1 out of every 2 flips in the real world does not affect theoretical probability. The theoretical probability of an event remains the same no matter how events turn out in the real world.

Let’s look at an example.

Find the probability of tossing a number cube and getting a 4.

First, find the total number of outcomes

Outcomes: 1, 2, 3, 4, 5, and 6

Total outcomes: 6

Second, find the number of favorable outcomes.

Favorable outcomes:

\begin{align*}\text{Getting a } 4=1 \text{ favorable outcome}\end{align*}

Then, find the ratio of favorable outcomes to total outcomes.

\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(4) &=& 1:6 \end{array}\end{align*}

A prediction is a reasonable guess about what will happen in the future. Good predictions should be based on facts and probability.

Predictions based on theoretical probability. These are the most reliable types of predictions, based on physical relationships that are easy to see and measure and that do not change over time. They include such things as:

• coin flips
• spinners
• number cubes

Examples

Example 1

Earlier, you were given a problem about Jessie and the spinner.

The spinner can be used to calculate theoretical probability when it is considered without an experiment. Looking at the spinner itself, you can calculate the chances of spinning one or more colors. You haven’t done an experiment with the spinner yet, you are simply calculating the probability by looking at the spinner.

The spinner has four colors so the total number of outcomes is 4. If she wanted to know the theoretical probability of spinning and getting on blue it would be 1:4 since there is only one section that is blue.

Experimental probability can be calculated once the spinner has been spun once or a number of times.

Example 2

Two number cubes are thrown. What is the theoretical probability of rolling a number greater than 8?

First, let’s think about the different combinations that you can possibly roll on the number cubes. If you roll a 1 and a 2 for instance, you get a sum of 3. Look at the table below to show all of the combinations.

The total number of outcomes is 36.

Next, the favorable outcomes are shaded in yellow in the table above.

\begin{align*}\# \ \text{favorable outcomes }= 10\end{align*}

Then, find the ratio of favorable outcomes to total outcomes.

\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(>8) &=& 10:36 \end{array}\end{align*}

Then, simplify the ratio.

\begin{align*}\begin{array}{rcl} P(>8) &=& 10:36 \\ P(>8) &=& 5:18 \end{array}\end{align*}

The theoretical probability of rolling a sum greater than 8 when rolling two number cubes is 5:18.

Example 3

What is the probability of tossing a number cube and having it come up a two or a three?

First, find the total number of outcomes

Outcomes: 1, 2, 3, 4, 5, and 6

Total outcomes: 6

Next, find the number of favorable outcomes.

Favorable outcomes:

\begin{align*}\text{Getting a }2 \text{ or a }3 = 2 \text{ favorable outcomes}\end{align*}

Then, find the ratio of favorable outcomes to total outcomes.

\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(2 \ \text{or} \ 3) &=& 2:6 \\ P(2 \ \text{or} \ 3) &=& 1:3 \end{array}\end{align*}

The theoretical probability of rolling a 2 or a 3 on a number cube is 1:3.

Example 4

What is the probability of tossing a number cube and having it come up even?

First, find the total number of outcomes

Outcomes: 1, 2, 3, 4, 5, and 6

Total outcomes: 6

Next, find the number of favorable outcomes.

Favorable outcomes:

\begin{align*}\begin{array}{rcl} \text{Getting an even number} &=& 2, 4, 6\\ &=& 3 \text{ favorable outcomes} \end{array}\end{align*}

Then, find the ratio of favorable outcomes to total outcomes.

\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(\text{event}\ \#) &=& 3:6 \\ P(\text{event}\ \#) &=& 1:2 \end{array}\end{align*}

The theoretical probability of rolling an even number on a number cube is 1:2.

Example 5

What is the probability of tossing a number cube and having it come up less than 6?

First, find the total number of outcomes

Outcomes: 1, 2, 3, 4, 5, and 6

Total outcomes: 6

Second, find the number of favorable outcomes.

Favorable outcomes:

\begin{align*}\begin{array}{rcl} \text{A number less than }6 &=& 1, 2, 3, 4, 5\\ &=& 5 \text{ favorable outcomes} \end{array}\end{align*}

Then, find the ratio of favorable outcomes to total outcomes.

\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(\# < 6) &=& 5:6 \end{array}\end{align*}

The theoretical probability of rolling a number less than 6 on a number cube is 5:6.

Review

Solve each problem.

A spinner has five sections: purple, yellow, green, blue and red.

1. Find the probability for the arrow landing on blue on the spinner:

1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Write the total number of outcomes.
4. Write the probability.

2. Find the probability for the arrow landing on red or green on the spinner:

1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Write the total number of outcomes.
4. Write the probability.

3. Find the probability for the arrow NOT landing on yellow on the spinner:

1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Write the total number of outcomes.
4. Write the probability.

4. Find the probability for rolling a 3 or 4 on the number cube:

1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Write the total number of outcomes.
4. Write the probability.

5. Find the probability for rolling greater than 2 on the number cube:

1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Write the total number of outcomes.
4. Write the probability.

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Color Highlighted Text Notes

Vocabulary Language: English

experimental probability

Experimental (empirical) probability is the actual probability of an event resulting from an experiment.

Favorable Outcome

A favorable outcome is the outcome that you are looking for in an experiment.

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.

Simulation

A way of collecting data using objects such as spinners, coins or cards.

Theoretical Probability

The theoretical probability of an event is the likelihood that the event will occur. It is calculated by finding the ratio of the number of favorable outcomes to the number of total outcomes.