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# Theoretical and Experimental Probability

## Ratio of successes to sum of successes and failures

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Experimental Probability

Jordan did a survey for his media class where he timed the length of TV advertisements (in seconds). The data he collected is shown below in the table.

 Length (s) Frequency 0 – 19 17 20 – 39 38 40 – 59 19 60+ 4

He needs to find the probability ratio that:

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

### Experimental Probability

Experimental probability is probability based on doing actual experiments -flipping coins, spinning spinners, picking ping pong balls out of a jar, and so on. To compute the experimental probability of a number cube landing on 3, you would need to conduct an experiment. Suppose you were to toss the number cube 60 times.

Favorable outcomes=rolling a 3\begin{align*}\text{Favorable outcomes} = \text{rolling a } 3 \end{align*}

Total outcomes=60 tosses\begin{align*}\text{Total outcomes} = 60 \ \text{tosses} \end{align*}

Experimental probability:

P(event)P(3)==# of favourable outcomestotal # of outcomes# of 3s tossedtotal # of tosses\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \frac{\# \ \text{of favourable outcomes}}{\text{total} \ \# \ \text{of outcomes}} \\ P(3) &=& \frac{\# \ \text{of} \ 3’s \ \text{tossed}}{\text{total} \ \# \ \text{of tosses}} \end{array}\end{align*}

Let’s look at an example.

What is the experimental probability of having the number cube land on 3 when the cube is rolled 60 times?

 Trial 1 2 3 4 5 6 Total Raw data: 3’s \begin{align*}|\end{align*} \begin{align*}|||\end{align*} \begin{align*}|\end{align*} \begin{align*}||\end{align*} \begin{align*}||\end{align*} Favorable outcomes: 3’s 1 3 0 1 2 2 9 Total Tosses 60

The data from the experiment shows that 3 turned up on the number cube 9 out of 60 times.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 9: 60\end{align*}

Simplified, this ratio becomes:

\begin{align*}\text{Experimental Probability} = 3: 20\end{align*}

You can see that it is only possible to calculate the experimental probability when you are actually doing experiments and counting results.

### Examples

#### Example 1

Jordan collected his data in a graph and needs to find the probability ratio that:

 Length (s) Frequency 0 – 19 17 20 – 39 38 40 – 59 19 60+ 4 Total 78

First, look at the data from the experiment to see how many times an advertisement was between 20 and 39 seconds. These are the favorable outcomes.

\begin{align*}\text{Favorable outcomes} = 38 \end{align*}

\begin{align*}\text{Total Outcomes} = 78\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 38: 78\end{align*}

Then, simplify the ratio.

\begin{align*}\text{Experimental Probability} = 38: 78\end{align*}

\begin{align*}\text{Experimental Probability} = 19: 39\end{align*}

The experimental probability that an advertisement will be between 20 and 39 seconds is 19:39.

First, look at the data from the experiment to see how many times an advertisement was less than 40 seconds. These are the favorable outcomes.

\begin{align*}\text{Favorable outcomes} = 17 + 38 = 55 \end{align*}

\begin{align*}\text{Total Outcomes} = 78\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 55: 78\end{align*}

The experimental probability that an advertisement will be less than 40 seconds is 55:78.

#### Example 2

Use the table to compute the experimental probability of a number cube landing on 6.

 Trial 1 2 3 4 5 Total Raw data: 6’s \begin{align*}||||\end{align*} \begin{align*}|\end{align*} \begin{align*}|\end{align*} \begin{align*}||\end{align*} \begin{align*}|\end{align*} Favorable outcomes: 6’s 4 1 1 2 1 9 Total Tosses 50

First, look at the data from the experiment to see how many times a six turned up when rolling a number cube and also what was the total number of outcomes.

\begin{align*}\text{Favorable outcomes} = 9\end{align*}

\begin{align*}\text{Total Outcomes}= 50\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 9: 50\end{align*}

The experimental probability is 9:50.

#### Example 3

What is the probability that the number would be a 2?

First, look at the data from the experiment to see how many times a 2 turned up when rolling a number cube and also what was the total number of outcomes.

\begin{align*}\text{Favorable outcomes} = 3\end{align*}

\begin{align*}\text{Total Outcomes} = 20\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 3: 20\end{align*}

The experimental probability is 3:20.

#### Example 4

What is the probability that the number would be a 5?

First, look at the data from the experiment to see how many times a 5 turned up when rolling a number cube and also what was the total number of outcomes.

\begin{align*}\text{Favorable outcomes} = 6\end{align*}

\begin{align*}\text{Total Outcomes} = 20\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 6: 20\end{align*}

Then, simplify the ratio.

\begin{align*}\text{Experimental Probability} = 6: 20\end{align*}

\begin{align*}\text{Experimental Probability} = 3: 10\end{align*}

The experimental probability is 3:10.

#### Example 5

What is the probability of not rolling a 5?

First, look at the data from the experiment to see how many times a 5 did not turn up when rolling a number cube and also what was the total number of outcomes. You know that it 5 did turn up 6 times in the 20 tosses.

\begin{align*}\text{Favorable outcomes} = 20 - 6 =14\end{align*}

\begin{align*}\text{Total Outcomes} = 20\end{align*}

Next, calculate the experimental probability.

\begin{align*}\text{Experimental Probability} = \text{favorable outcomes}: \text{total outcomes}\end{align*}

\begin{align*}\text{Experimental Probability} = 14: 20\end{align*}

Then, simplify the ratio.

\begin{align*}\text{Experimental Probability} = 14:20\end{align*}

\begin{align*}\text{Experimental Probability} = 7:10\end{align*}

The experimental probability is 7:10.

### Review

Find the probability for rolling less than 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 1 or 6 on the number cube.

6. List each favorable outcome.

7. Count the number of favorable outcomes.

8. Write the total number of outcomes.

9. Write the probability.

10. A box contains 12 slips of paper numbered 1 to 12. Find the probability for randomly choosing a slip with a number less than 4 on it.

11. List each favorable outcome.

12. Count the number of favorable outcomes.

13. Write the total number of outcomes.

14. Write the probability.

Use the table to answer the questions. Express all ratios in simplest form.

Use the table to compute the experimental probability of flipping a coin and having it land on heads.

 Trial 1 2 3 4 5 6 Total Raw data(heads) \begin{align*}\bcancel{||||}\end{align*} \begin{align*}\bcancel{||||} \ |\end{align*} \begin{align*}\bcancel{||||} \ |\end{align*} \begin{align*}|||\end{align*} \begin{align*}\bcancel{||||} \ |\end{align*} \begin{align*}\bcancel{||||}\end{align*} Number of heads 5 6 6 3 6 5 31 Total number of flips 10 10 10 10 10 10 60

15. How many favorable outcomes were there in the experiment?

16. How many total outcomes were there in the experiment?

17. What was the experimental probability of the coin landing on heads?

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Color Highlighted Text Notes

### Vocabulary Language: English

Event

An event is a set of one or more possible results of a probability experiment.

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.

Outcome

An outcome of a probability experiment is one possible end result.

Probability

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

theoretical probability

Theoretical probability is the probability ration of the number of favourable outcomes divided by the number of possible outcomes.

trial

A trial is one “run” of a particular experiment.