Brian is planning to have his birthday party this year at the local skate park. He is a little nervous about the weather, fearing his party might get rained out. He checks out the past weather history for the date of his party. In the past ten years, there has only been rain in the area of the skate park twice on the date of the party. He feels more confident, knowing that the probability is high that there will not be rain. What is the probability of a rain-free birthday at the skate park for Brian's birthday party?

In this concept, you will learn you how to figure out probability by thinking about favorable outcomes and total outcomes.

### Defining Probability

Probability is something that you hear about all the time. Anytime you talk about the chances that something will or won’t happen, you are talking about probability. The trick about probability is that it isn’t just about talking. It is also about math. There are mathematical ways of figuring out the likelihood that an event is going to or not going to occur.

**Probability** is the likelihood that an event will occur. It is a mathematical way of calculating how likely an event is likely to occur. An **event** is a result of an experiment or activity that might include such things as:

- flipping a coin
- spinning a spinner
- rolling a number cube
- choosing an item from a jar or bag

Probability is calculated by looking at the ratio of **favorable outcomes** to **total outcomes** in a given situation. In ratio form, the probability of an event is:

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

An **outcome** is a possible result of some event occurring. For example, when you flip a coin, “heads” is one outcome; tails is a second outcome. **Total outcomes** are computed simply by counting all possible outcomes.

Keep in mind as you go through this concept that all outcomes used are presumed to be “fair.” For example, when you flip a coin, the outcomes of heads or tails are equally likely. When you spin a spinner, sections are all of equal size and equally likely to be landed on. When you toss a number cube, faces of the cube are the same size and again equally likely to be landed on. And so on.

For flipping a coin:

\begin{align*}\text{total outcomes} & = \text{heads, tails}\\ & = 2 \ \text{total outcomes}\end{align*}

For tossing a dice:

\begin{align*}\text{total outcomes} & = \cdot \ 1 \ \cdot \cdot \ 2 \ \cdot \cdot \cdot \ 3 \ \cdot \cdot \cdot \cdot \ 4 \ \cdot \cdot \cdot \cdot \cdot \ 5 \ \cdot \cdot \cdot \cdot \cdot \cdot \ 6\\ & = 6 \ \text{total outcomes}\end{align*}

For selecting a day of the week:

\begin{align*}\text{total outcomes} & = \text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\\ & = 7 \ \text{total outcomes}\end{align*}

**Favorable outcomes** are the specific outcomes you are looking for.

For flipping a coin and having it come up heads:

\begin{align*}\text{favorable outcomes} & = \text{heads, tails}\\ & = 1 \ \text{favorable outcome}\end{align*}

For tossing a number cube and having it come with up an even number:

\begin{align*}\text{favorable outcomes} & = \cdot \ 1 \ \cdot \cdot \ 2 \ \cdot \cdot \cdot \ 3 \ \cdot \cdot \cdot \cdot \ 4 \ \cdot \cdot \cdot \cdot \cdot \ 5 \ \cdot \cdot \cdot \cdot \cdot \cdot \ 6\\ & = 3 \ \text{favorable outcomes}\end{align*}

For randomly choosing a date and have it land on a weekday:

\begin{align*}\text{favorable outcomes} & = \text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\\ & = 5 \ \text{favorable outcomes}\end{align*}

To write a ratio, the favorable outcome is compared to the total outcomes. Comparing favorable outcomes to possible total outcomes is what is called **theoretical probability.**

The probability of any event is written as \begin{align*}P (\text{event})\end{align*}.

For example:

\begin{align*}P(A)\end{align*} is the probability that *event* \begin{align*}A\end{align*} will occur.

\begin{align*}P (\text{heads})\end{align*} is the probability that *heads* will turn up on a flipped coin.

\begin{align*}P (5)\end{align*} is the probability that a number cube will turn up as *5.*

Here are two examples:

What is the probability of flipping heads on a coin?

\begin{align*} \text {favorable outcome}\ =1 \text { since there is one head on a coin}\end{align*}

\begin{align*}\text {Total outcomes}\ = 2 \text { since there is the possibility of heads or tails}\end{align*}

The answer is 1:2.

For tossing a number cube and having it land an even number:

The final answer is 1:2.

Any time the answer can be simplified, you should do so.

To find probability, follow the steps below:

What is the probability of the arrow landing on a yellow section?

First, count the number of favorable outcomes.

There are 2 yellow spaces, so favorable outcomes = 2

Next, count the number of total outcomes.

There are 5 spaces in all, so total outcomes = 5

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

\begin{align*}P (\text{yellow}) &= \text{favorable outcomes} : \text{total outcomes}\ = 2:5\end{align*}The answer is 2:5.

### Examples

#### Example 1

Earlier, you were given a problem about Brian's worry about rain at his skate park birthday party.

What is the probability that he will escape rain at his birthday party if there have been 2 days of rain at the skate park on the same date over the past 10 years?

First, count the number of favorable outcomes.

If there have been 2 days without rain over the past 10 years, there have been 8 days without rain which is the favorable outcome. So favorable outcomes = 8

Next, count the number of total outcomes.

There are 10 days in all, so total outcomes = 10

Then, write the ratio of favorable outcomes to total outcomes, making sure to simplify if possible.

\begin{align*}P (\text{no rain}) &= \text{favorable outcomes} : \text{total outcomes}\ = 8:10 = 4:5\end{align*}The answer is 4:5.

#### Example 2

What is the probability of the arrow landing on a silver or pink section when using the spinner above?

First, count the number of favorable outcomes.

There are 2 silver spaces and 1 pink space, so favorable outcomes = 3

Next, count the number of total outcomes.

There are 5 spaces in all, so total outcomes = 5

Then, write the ratio of favorable outcomes to total outcomes

\begin{align*}P (\text{silver or pink}) = \text{favorable outcomes} : \text{total outcomes} = 3:5\end{align*} The answer is 3:5.

#### Example 3

What is the probability of rolling a 1 or a 3 on a number cube?

First, count the number of favorable outcomes.

There is one 1 and one 3 on a number cube, so favorable outcomes = 2

Next, count the number of total outcomes.There are 6 numbers on a number cube in all, so total outcomes = 6

Then, write the ratio of favorable outcomes to total outcomes, making sure to simply if possible.

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

The answer is 1:3.

#### Example 4

If there are four blue marbles and one red marble in a bag, what is the probability of pulling out a red one?

First, count the number of favorable outcomes.

There is one red marble in the bag, so favorable outcomes = 1

Next, count the number of total outcomes.

There are 5 marbles in all, so total outcomes = 5

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

\begin{align*}P (\text{red}) &= \text{favorable outcomes} : \text{total outcomes}\ = 1:5\end{align*}The answer is 1:5.

#### Example 5

What is the probability of pulling out a blue marble if there are four blue marbles and one red marble in a bag?

First, count the number of favorable outcomes.

There are 4 blue marbles, so favorable outcomes = 4

Next, count the number of total outcomes.

There are 5 marbles in all, so total outcomes = 5

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

\begin{align*}P (\text{blue}) &= \text{favorable outcomes} : \text{total outcomes}\ = 4:5\end{align*}The answer is 4:5.

### Review

Answer each question or solve each problem as it connects to probability.

For rolling a 4 on the number cube.

- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.

For rolling a number greater than 2 on the number cube:

- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.

For rolling a 5 or 6 on a number cube:

- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.

A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with an even number on it:

- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.

A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with a number greater than 3:

- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be orange.

- Count the number of favorable outcomes.
- Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be large:

- Count the number of favorable outcomes.
- Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be blue:

- Count the number of favorable outcomes.
- Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be small:

- Count the number of favorable outcomes.
- Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be orange and large:

- Count the number of favorable outcomes.
- Write the total number of outcomes.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.1.