## Probability: Overview

When an official tosses a coin in the air at the beginning of a football game and one of the team captains calls “heads” or “tails,” what is the probability of the flipped coin coming up heads? Probability can be defined as “the likelihood that an event will occur.” An event is a specific outcome. In this case, the event is the coin landing on heads. The reason a coin is tossed before football games, for example, is because it allows a fair result, with two equally likely outcomes, heads or tails. The team captain will either win or lose the toss. The probability that the coin will come up heads is 1 out of 2—one outcome, heads, out of two possible outcomes, heads or tails. This probability can be expressed as a ratio, , and it can be stated as an equation: P(heads) = . Thus, probability can be expressed as the ratio of the number of favorable outcomes to the number of possible outcomes.

To be able to determine the probability of an event, it is necessary to know all the possible outcomes for the event.

One way to identify all the possible outcomes of an event is to list all possible choices in an organized way. For example, if there are 4 flavors of ice cream—chocolate, vanilla, strawberry, and mint—and 2 types of cones, sugar and waffle, how many different choices of ice-cream cones are possible?

List of Ice-Cream Cone Choices
Sugar cone, chocolate ice cream
Sugar cone, vanilla ice cream
Sugar cone, strawberry ice cream
Sugar cone, mint ice cream
Waffle cone, chocolate ice cream
Waffle cone, vanilla ice cream
Waffle cone, strawberry ice cream
Waffle cone, mint ice cream

When making a list to find all possible choices, students need to be careful not to duplicate choices.

A tree diagram is another way to show all possible choices.

The tree diagram shows us more clearly the total number of choices. Two cones and 4 flavors result in 8 choices. Multiplication is another way to find the total number of choices: 2 x 4 = 8. Two choices in the first group (cones) and 4 choices in the second group (flavors) give 8 choices of cones and flavors. Multiplying the choices in each group will always give us the total number of possible choices.

Continuing with the ice-cream example, suppose you selected an ice-cream cone from a box without looking. What is the probability that you would select a waffle cone with chocolate ice cream? Students should recognize that the result of the tree diagram, organized list, or multiplication gives them the total number of possible choices (8). The probability that you will choose a waffle cone with chocolate ice cream is 1 out of 8, or .

When you describe an event, you can be specific about the likelihood of its occurring. Spin the spinner pictured below. Note that it has eight sections of equal size.

The probability of spinning and landing on a number greater than 13 is 0, because it is impossible to land on a number that doesn't appear on this spinner. That event has a probability of 0. A probability of 1 means the event is certain to happen, such as spinning and landing on a number less than 9. The closer a probability is to 1, the more likely an event is going to occur. So the probability that an event will happen ranges from 0 to 1. On this spinner, landing on a number between 1 and 8 is more likely to happen than landing on a 3.

You can show the probability of an event with a number line.

Using the above spinner you can locate the probability of an event on the number line. The probability of spinning and landing on 5 is 1 out of 8, or . This event is unlikely to happen. The probability of landing on an even number is 4 out of 8, or . You are equally likely to land on an even number as an odd number. The probability of landing on a number greater than 1 is 7 out of 8, or . This event is likely to occur.

When we discuss probability we are describing the likelihood that an event will occur. We can always express this likelihood as a fraction, since we are talking about a ratio of the number of favorable outcomes to the number of possible outcomes.

Using a deck of 48 number cards containing 4 each of cards numbered 1-12, what is the probability of picking a 9 at random? Since all 48 cards are of equal size, there are 48 possible outcomes of choosing 1 card. There are four 9s in the deck. The probability can be written like this.

The probability of choosing a 9 is written as the fraction . Probabilities should be expressed as fractions in simplest form.

Students will already be familiar with rolling a number cube, and you can use this model to reinforce probability concepts. When Carl rolls a number cube numbered 1, 2, 3, 4, 5, and 6, what is the probability he will roll a 5? Rolling the cube has 6 equally likely outcomes. Having the 5 land faceup is one of 6 possible outcomes. The probability of rolling a 5 is 1 out of 6, or . The probability of rolling an odd number is because of the 6 possible outcomes, 3 are odd numbers, 1, 3, or 5. Then, = . Since 7 is not one of the choices, the probability of Carl rolling a 7 is 0.

Probability is often used to make predictions about possible outcomes based on data. Students need to be able to apply what they learn about probability to real-life situations and problems.

Example: The science club was sponsoring an after-school dance and wanted to sell snow cones to raise money. They could only afford to order two flavors of syrup and wanted to predict which two flavors would sell the best. They conducted a lunchroom survey of 50 students and compiled the following data.

Flavor Students' Choices
Cherry 18
Lime 10
Grape  3
Raspberry  4
Orange 15

If the science club orders cherry and orange syrup for the dance, they predict that the probability of having a favorite flavor will be 33 out of 50, or . So, these two flavors would more than likely seem the best.