11.4 Theoretical and Experimental Probability

Learning Objectives

I can define theoretical and experimental probability.
I can describe sample spaces.
I can conduct probability simulations.

Introduction

Suppose the numbers 0 through 9 were each written on a piece of paper and placed in a bag. If one of the pieces of paper were chosen at random, what is the theoretical probability that it would be an even number? If this experiment were repeated 10 times, and an even number were chosen 6 times, what would be the experimental probability of choosing an even number? In this Concept, you'll learn all about theoretical and experimental probability, including sample spaces and probability simulations, so that you can answer questions like these.

Guided Learning

So far in this unit, you have solved problems dealing with possible situations. In this lesson, we will begin to compare possible events with results from actual probability experiments.

Almost all companies use some form of probability. Automotive companies want to determine the likelihood of their new vehicle being a big seller. Cereal manufacturers want to know the probability that their cereal will sell more than the competition. Pharmaceutical corporations need to know the likelihood of a new drug harming those who take it. Even politicians want to know the probability of receiving enough votes to win the election.

Probabilities can start with an experiment. An experiment is a controlled study. For example, suppose you want to know if the probability of getting tails when flipping a coin is actually $\frac{1}{2}$ . By randomly grabbing a penny and making a tally chart of heads and tails, you are performing an experiment.

The set of all possible outcomes is called the sample space of the experiment.

If you were to toss a coin, there are only possible outcomes: heads or tails. That means the sample space for tossing a coin is {heads, tails}.

List the sample space for rolling a regular die.

Solution:

A die is a six-sided figure with dots representing the numbers one through six. So the sample space is all the possible outcomes (i.e., what numbers could possibly be rolled).

$S=\left \{1,2,3,4,5,6,\right \}$

Once you have determined the number of items in the sample space, you can compute the probability of a particular event.

Any one possible outcome of the experiment is called an event.

Theoretical probability is a ratio expressing the ways to be successful to the total events in an experiment. This probability ratio will show what we expect to happen based on the information we have about the number of ways to be successful and the possible outcomes. A shorter way to write this is:

$\text{Probability} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{total \ number \ of \ possible \ outcomes}$

We have seen this expressed as

Both formulas are displaying the same concept: the likelihood that you will be successful in achieving an outcome. This likelihood is called the theoretical probability, our prediction at an outcome. It is possible to have a 0% probability (event can not happen) and a 100% probability (the event will always happen). Probabilities can be expressed three ways:

As a fraction
As a percent
As a decimal

Example A

Determine the theoretical probability of rolling a five on a die.

Solution: There are six events in the sample space. There is one way to roll a five.

$P(rolling \ a \ 5)=\frac{1}{6} \approx 16.67\%$

Example B

Suppose you want to know the theoretical probability of getting a head or a tail when flipping a coin. There would be two ways of obtaining a success (heads or tails) and two possible outcomes (heads or tails).

$P(success)=\frac{2}{2}=1$

This is a very important concept of probability. It can be shown by calculating the probability of each event and then finding the sum of the events.

The probability of flipping heads is P = $\frac{1}{2}$ . The probability of flipping tails is P = $\frac{1}{2}$ .

The sum of all probabilities is $\frac{1}{2}$ + $\frac{1}{2}$ = 1

The sum of all the individual event probabilities is 100%, or 1.

Conducting an Experiment

Conducting an experiment for probability purposes is also called probability simulation. Suppose you wanted to conduct the coin experiment in this concept's opener. By grabbing a random coin, flipping it, and recording what comes up is a probability simulation.

Using this information, you can determine the experimental probability of tossing a coin and seeing a tail on its landing. The experimental probability is the ratio of the proposed outcome to the number of experiment trials. Experimental probability shows the results of the actual experiment.

$P(success)= \frac{number \ of \ times \ the \ event \ occurred}{total \ number \ of \ trials \ of \ experiment}$

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Take a minute to write the theoretical and experimental probability ratios.

Example C

Compare the theoretical probability of flipping a tail on a coin to the experimental probability of flipping a tail.

Chenoa conducted a probability simulation by flipping a coin 10 times and recording the results. She flipped four heads and six tails.

Theoretical Probability

There are two events in the sample space. There is one way to flip a tail, so the theoretical probability is:

$P(flipping \ a \ tail)=\frac{1}{2}$

The theoretical probability told Chenoa to expect to flip tails half or 50% of the time.

Experimental Probability

The coin toss simulation Chenoa performed stated there were six tails out of ten tosses. Her experimental probability is:

$P(flipping \ a \ tail)=\frac{6}{10}$

Chenoa's experimental probability ended up being 60%. The experimental probability (60%) in this case is greater than the theoretical probability (50%).

It is often the case that the theoretical and experimental probabilities do not match. Remember that when we calculate theoretical probability, we are calculating what we expect to happen. Conducting an experiment of random events will often not match what we expect, but results are usually close to the expected.

Finding Odds For and Against

People often hear about the odds of an event occurring. Odds are similar to probability with the exception of the ratio's denominator. The odds in favor of an event is the ratio of the number of successful events to the number of non-successful events. The odds of an event compares the number of ways to get success with the number of ways to not get success.

$\text{Odds} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{number \ of \ ways \ to \ not \ get \ success}$

Calculating odds requires that you know the sample space and what outcomes will achieve success and what possible outcomes will not be successful.

Example D

Suppose we were interested in the odds of rolling a 5 on a die.

The odds of rolling a 5 on a die is $\frac{1}{5}$ .

What if we were interested in determining the odds against rolling a 5 on a die? Success in this case would mean rolling anything except the 5. There are five outcomes other than a “5” and one outcome of a “5.”

$Odds \ against \ rolling \ a \ 5= \frac{5}{1}$

Notice the “odds against” ratio is the reciprocal of the “odds in favor” ratio.

Example E

Find the odds against rolling a number larger than 2 on a standard die.

Solution:

Success for this problem can be identified as rolling 2 or less, meaning 1 or 2 on the die. There are four outcomes on a standard die larger than $2: \left \{3,4,5,6\right \}$ .

$Odds \ against \ rolling>2= \frac{2}{4}$

Guided Practice

1. What is the theoretical probability of rolling a four using a regular six sided die?

2. What are the odds in favor of rolling a four using a regular six sided die?

3. What are the odds against rolling a five using a regular six sided die?

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Check your work with a partner.

Solutions

1. ⅙

2. ⅕

3. 5/1

Review

Theoretical probability

Theoretical probability is a ratio expressing the ways to be successful to the total events in an experiment.

$\text{Probability} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{total \ number \ of \ possible \ outcomes}$

Experimental probability

The ratio of the proposed outcome to the number of experiment trials is called the experimental probability.

$P(success)= \frac{number \ of \ times \ the \ event \ occurred}{total \ number \ of \ trials \ of \ experiment}$

Odds in favor of an event

The ratio of the number of successful events to the number of non-successful events is called the odds of an event.

$\text{Odds} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{number \ of \ ways \ to \ not \ get \ success}$

Experiment

An experiment is a controlled study.

Sample space

The set of all possible outcomes is called the sample space of the experiment.

Probability simulation

Conducting an experiment for probability purposes is also called probability simulation.

Event

Any one possible outcome of the experiment is called an event.

Additional Resources

Theoretical and Experimental Probability Video

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