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# Theoretical and Experimental Probability

## Ratio of successes to sum of successes and failures

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Practice Theoretical and Experimental Probability
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Experimental Probability

Have you ever wanted to trade jobs with someone else? Take a look at this dilemma between two friends.

“I don’t want to work on the weekend,” Carey said to Telly at lunch one day.

“But that was part of the deal. We both have to work one day out of the weekend,” Telly said.

“Well, which day do you want?” Carey asked.

“I don’t know. I haven’t really thought about it,” Telly said. “But we could make it really random.”

Telly took two pieces of paper and wrote Saturday on one and Sunday on the other.

“Now we can figure out the probability of you getting Saturday or Sunday,” she said.

We can stop there. This Concept is all about probability. Telly’s experiment is an example of experimental probability. Let’s talk more about this at the end of the Concept.

### Guidance

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 the number cube landing on 3 you would need to conduct an experiment. Suppose you were to toss the number cube 60 times.

Favorable outcomes :

Total outcomes : 60 tosses

Experimental probability :

$P(3) =\frac{favorable \ outcomes}{total \ outcomes}=\frac{Number \ of \ 3's}{Total \ Number \ of \ tosses}$

Write this comparison down in your notebooks.

Take a look at this situation.

What is the experimental probability of having the number cube land on 3?

trial 1 2 3 4 5 6 Total
raw data:3s ${|}$ ${|||}$ ${|}$ ${||}$ ${||}$
favorable outcomes:3s 1 3 0 1 2 2 9
total tosses total outcomes 10 10 10 10 10 10 60
experimental probability: favorable outcomes to total outcomes x x x x x x $9:60=3:20$

The data from the experiment shows that 3 turned up on the number cube 9 out of 60 times. Simplified, this ratio becomes:

$\text{Favorable outcomes}:\text{total outcomes}= 3:20$

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

A number cube was tossed twenty times. The number 2 came up 3 times and the number 5 came up six times. Use this information to answer the following questions.

#### Example A

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

Solution: $2:20$ or $1:10$

#### Example B

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

Solution: $6:20$ or $3:10$

#### Example C

What is the probability of not rolling a 5?

Solution: $7:10$

Now let's go back to the dilemma from the beginning of the Concept.

Carey has a chance of working Saturday or Sunday. There are two possible outcomes. She has a one out of 2 chance of working on Saturday and a one out of two chance of working on Sunday.

$\frac{1}{2}$

.50

50% chance or probability for each outcome.

### Guided Practice

Here is one for you to try on your own.

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

trial 1 2 3 4 5 Total
raw data ${||||}$ ${|}$ ${|}$ ${||}$ ${|}$ x
number of 6's 4 1 1 2 1 9
total tosses 10 10 10 10 10 50

Solution

You can see from the experiment that the number cube was tossed 50 times.

The total number of sixes to appear during this experiment was 9.

The experimental probability of rolling a 6 is $9:50$ .

### Explore More

Directions: 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.

Directions: 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) $\cancel{||||}$ $\cancel{||||} \ {|}$ $\cancel{||||} \ {|}$ ${|||}$ $\cancel{||||} \ {|}$ $\cancel{||||}$
number of heads 5 6 6 3 6 5 31
total number of flips 10 10 10 10 10 10 60
1. How many favorable outcomes were there in the experiment?
2. How many total outcomes were there in the experiment?
3. What was the experimental probability of the coin landing on heads?

### Vocabulary Language: English

Event

Event

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

experimental probability

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

Favorable Outcome

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

Outcome

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

Probability

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

theoretical probability

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

trial

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

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