Suppose the numbers 0 through 9 were each written on a piece of paper and placed in a bag. If one piece of paper is chosen at random, what is the theoretical probability that it is an even number? If this experiment is repeated 10 times, and an even number is chosen 6 times, what is the experimental probability of choosing an even number?

### Probability

Previously we worked on solving problems dealing with definite situations. In this concept, we will begin to look at a branch of mathematics that deals with **possible situations**. The study of probability involves applying formulas to determine the percentage chance of an event occurring.

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**. As you learned in a previous concept, 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 \begin{align*}\frac{1}{2}\end{align*}. 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.

The sample space for tossing a coin is \begin{align*}\left \{heads,\ tails\right \}\end{align*}.

#### Let's list the sample space for rolling a die:

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).

\begin{align*}S=\left \{1,2,3,4,5,6,\right \}\end{align*}

#### Theoretical Probability

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. A shorter way to write this is:

\begin{align*}\text{Probability} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{total \ number \ of \ possible \ outcomes}\end{align*}

Probabilities are expressed three ways:

- As a fraction
- As a percent
- As a decimal

Suppose you wanted to know the probability of getting a head or a tail when flipping a coin.

There would be two ways of obtaining a success and two possible outcomes.

\begin{align*}P(success)=\frac{2}{2}=1\end{align*}

This is a very important concept of probability.

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

#### Let's determine the theoretical probability of rolling a five on a die:

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

\begin{align*}P(rolling \ a \ 5)=\frac{1}{6} \approx 16.67\%\end{align*}

#### Conducting an Experiment

Conducting an experiment for probability purposes is also called probability simulation. Suppose you wanted to conduct an experiment to determine the probability of getting tails when tossing a coin 10 times. By grabbing a random coin, flipping it, and recording what comes up is a probability simulation. You can also simulate an experiment using a graphing calculator application.

**Performing an Experiment Using the TI-84 Graphing Calculator**

There is an application on the TI calculators called the coin toss. Among others (including the dice roll, spinners, and picking random numbers), the coin toss is an excellent application for when you what to find the probabilities for a coin tossed more than four times or more than one coin being tossed multiple times.

Let’s say you want to see one coin being tossed one time. Here is what the calculator will show and the key strokes to get to this toss.

Let’s say you want to see one coin being tossed ten times. Here is what the calculator will show and the key strokes to get to this sequence. Try it on your own.

We can actually see how many heads and tails occurred in the tossing of the 10 coins. If you click on the right arrow \begin{align*}(>)\end{align*} the frequency label will show you how many of the tosses came up heads.

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.

\begin{align*}P(success)= \frac{number \ of \ times \ the \ event \ occurred}{total \ number \ of \ trials \ of \ experiment}\end{align*}

#### Let's compare the theoretical probability of flipping a tail on a coin to the experimental probability of flipping a tail using the graphing calculator information:

There are two events in the sample space. There is one way to flip a tail.

\begin{align*}P(flipping \ a \ tail)=\frac{1}{2}\end{align*}

The coin toss simulation the calculator performed stated there were six tails out of ten tosses.

\begin{align*}P(flipping \ a \ tail)=\frac{6}{10}\end{align*}

The experimental probability (60%) in this case is greater than the theoretical probability (50%).

#### Finding Odds For and Against

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.

\begin{align*}\text{Odds} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{number \ of \ ways \ to \ not \ get \ success}\end{align*}

#### Let's find the odds of rolling a 5 on a die:

What if we were interested in determining the odds against rolling a 5 on a die. There are five outcomes other than a “5” and one outcome of a “5.”

\begin{align*}Odds \ against \ rolling \ a \ 5= \frac{5}{1}\end{align*}

### Examples

#### Example 1

Earlier, you were told that the numbers 0 through 9 were written on a piece of paper placed in a bag and one piece of paper is drawn at random. What is the theoretical probability that it is an even number? If this experiment is repeated 10 times and an even number is chosen 6 times, what is the experimental probability of choosing an even number?

The theoretical probability of an event is the number of ways to get a success divided by the number of possible outcomes. In this case, there are 4 even numbers in 0 through 9 which is the result that we want. There are 10 pieces of paper in the bag since there are 10 numbers in 0 through 9.

\begin{align*}P(even)=\frac{4}{10}=\frac{2}{5}=40\%\end{align*}

The experimental probability is the number of times the event occurred divided by the total number of trials. If there are 10 trials, and an even number is chosen 6 times, then we have:

\begin{align*}P(even)=\frac{6}{10}=\frac{3}{5} = 60\%\end{align*}

The theoretical probability is 40% and the experimental probability is 60%.

#### Example 2

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

There are four outcomes on a standard die larger than \begin{align*}2: \left \{3,4,5,6\right \}\end{align*}.

\begin{align*}Odds \ against \ rolling>2= \frac{2}{4}\end{align*}

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

### Review

- Define experimental probability.
- How is experimental probability different from theoretical probability?
- Complete the table below, converting between probability values.

Fraction |
Decimal |
Percent |
---|---|---|

98% | ||

0.015 | ||

\begin{align*}\frac{1}{16}\end{align*} | ||

\begin{align*}\frac{2}{3}\end{align*} | ||

62% | ||

0.73 |

Use the “SPINNER” application in the Probability Simulator for the following questions. Set the spinner to five pieces.

- What is the sample space?
- Find the theoretical probability, \begin{align*}P\end{align*}(spinning \begin{align*}a\end{align*} 4).
- Conduct an experiment by spinning the spinner 15 times and recording each number the spinner lands on.
- What is the experimental probability, \begin{align*}P\end{align*}(spinning \begin{align*}a\end{align*} 4)?
- Give an event with a 0% probability.

In 9–18, use a standard 52-card deck to answer the questions.

- How many values are in the sample space? What could be an easy way to list all these values?
- Determine \begin{align*}P\end{align*}(King).
- What are the odds against drawing a face card?
- What are the odds in favor of drawing a six?
- Determine \begin{align*}P\end{align*}(Diamond).
- Determine \begin{align*}P\end{align*}(Nine of Clubs).
- Determine \begin{align*}P\end{align*}(King or 8 of Hearts).
- What are the odds against drawing a spade?
- What are the odds against drawing a red card?
- Give an event with 100% probability.
- Jorge says there is a 60% chance of rain tomorrow. Is this a strong likelihood? Explain your reasoning.
- What is the probability there will be a hurricane in your area tomorrow? Why did you choose this percentage?

Consider flipping two coins at the same time.

- Write the sample space.
- What is the probability of flipping one head and one tail?
- What is \begin{align*}P\end{align*}(both heads)?
- Give an event with 100% probability.
- Conduct the experiment 20 times. Find the experimental probability for flipping both heads. Is this different from the theoretical probability?

**Mixed Review**

- Graph the inequality on a coordinate plane: \begin{align*}-2 \le w<6\end{align*}.
- Solve and graph the solutions using a number line: \begin{align*}|n-3|>12\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.12.