Likelihood from 0 to 1 and Sample Spaces | Grade 7 Math | LevelKite

Think about one roll of a fair number cube. Its six faces are numbered $1$ to $6$ , and fair means each face has the same chance of landing on top. Rolling a $7$ can never happen. Rolling a number less than $7$ always happens. Rolling an even number may or may not happen. In this lesson, you will measure chances like these with a number.

Three terms help you talk about chance:

An experiment is an action whose result you cannot know ahead of time, such as one roll of the cube.
An outcome is one possible result of an experiment, such as rolling a $4$ .
An event is a result, or a group of results, that you ask about, such as rolling an even number. Each outcome either fits the event or does not. The outcomes $2$ , $4$ and $6$ fit this event. No outcome fits the event of rolling a $7$ .

The probability of an event is a number from $0$ to $1$ that measures its likelihood, which means how likely the event is to happen. You write the probability of an event as $P(\text{event})$ . The closer the number is to $1$ , the more likely the event. The closer it is to $0$ , the less likely.

Look again at one roll of the fair number cube:

Rolling a $7$ cannot happen, so its probability is $0$ .
Rolling a number less than $7$ must happen, so its probability is $1$ .
Rolling an even number: three faces are even, three are odd, and each face has the same chance. The event is just as likely to happen as not, so its probability is $\frac{1}{2}$ . You can also write this probability as the decimal $0.5$ or as 50%.

The number line below shows the scale. Its markers sit at $0$ , $0.5$ and $1$ : the two ends and the halfway point. The table after it gives a word for each part of the scale.

Rolling a

7

sits at

0

, rolling an even number sits at

0.5

, and rolling a number less than

7

sits at

1

.

Words for each part of the probability scale

Word	What it means	Probability
impossible	The event cannot happen.	$0$
unlikely	The event is less likely to happen than not.	greater than $0$ and less than $\frac{1}{2}$
as likely as not	The event is just as likely to happen as not.	$\frac{1}{2}$
likely	The event is more likely to happen than not.	greater than $\frac{1}{2}$ and less than $1$
certain	The event must happen.	$1$

Worked example

Describing chances from a forecast

A weather forecast gives the chance of rain as 90% for Friday, 50% for Saturday and 10% for Sunday. Describe how likely rain is on each day, using the scale from $0$ to $1$ .

Put each chance on the scale from 0 to 1. A percent is a number out of 100, so write each percent as a fraction over 100 and then as a decimal.
$\frac{90}{100} = 0.9 \qquad \frac{50}{100} = 0.5 \qquad \frac{10}{100} = 0.1$
Friday: 0.9 is greater than 0.5 but less than 1, so rain is more likely to happen than not, but not certain. That is what likely means.
$0.5 < 0.9 < 1$
Saturday: 0.5 is exactly one half, so rain is just as likely to happen as not.
$0.5 = \frac{1}{2}$
Sunday: 0.1 is greater than 0 but less than 0.5, so rain is less likely to happen than not, but not impossible. That is what unlikely means.
$0 < 0.1 < 0.5$
Compare the days. The number line after this example marks 0.1, 0.5 and 0.9. A marker farther to the right shows a more likely event, so rain is most likely on Friday and least likely on Sunday.
$0.1 < 0.5 < 0.9$

Answer

Rain is likely on Friday ( $0.9$ ), as likely as not on Saturday ( $0.5$ ) and unlikely on Sunday ( $0.1$ ).

Sunday's chance of rain is at

0.1

, Saturday's is at

0.5

and Friday's is at

0.9

.

In the forecast, someone gave you the probabilities. To judge an event yourself, you first need to know everything that can happen.

The sample space of an experiment is the set of all its possible outcomes: a complete list with each outcome written once. A simple event is an event from an experiment with a single action, such as one roll, one spin or one draw. To list the sample space for a simple event, list every possible result of that one action:

Decide what you record as the result, such as a number, a letter or a colour.
Write each different result once, with commas between them, inside braces (curly brackets).
Check that no possible result is missing and that none appears twice.

Worked example

Listing the sample space for one roll

You roll a fair number cube once. Its six faces are numbered $1$ to $6$ . List the sample space. Then list the outcomes that fit the event of rolling a number greater than $4$ .

Name the single action and the result you record. The action is one roll, and you record the number on the top face.
Write each possible number once, in counting order so that you skip none.
$\lbrace 1, 2, 3, 4, 5, 6 \rbrace$
Check the list. The cube has 6 faces with 6 different numbers, and the list has 6 different outcomes. Nothing is missing and nothing appears twice.
Pick out the outcomes that fit the event. In the sample space, only 5 and 6 are greater than 4.
$\lbrace 5, 6 \rbrace$

Answer

The sample space is $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$ . The outcomes $5$ and $6$ fit the event of rolling a number greater than $4$ .

Key concept

A scale for likelihood, a list of outcomes

The probability of an event is a number from $0$ to $1$ that tells how likely the event is. The closer it is to $1$ , the more likely the event.

The sample space lists every possible outcome of the experiment. Use it to place an event on the scale:

No outcome fits the event: the event is impossible, with probability $0$ .
Every outcome fits the event: the event is certain, with probability $1$ .
Compare counts only when the outcomes are equally likely. If exactly half of them fit the event, it is as likely as not, with probability $\frac{1}{2}$ . If some but fewer than half fit, it is unlikely. If more than half but not all fit, it is likely.

0 \le P(\text{event}) \le 1

Worked example

Using a sample space to describe likelihood

A spinner has $10$ equal-sized sections, and the pointer is equally likely to stop in any one of them. Five sections are labelled A, four are labelled B and one is labelled C. You spin once and record the letter. List the sample space. Then describe the likelihood of each event: landing on A; landing on C; landing on D; landing on A, B or C.

One spin is a single action, and the result you record is a letter. Write each different letter once, even though A and B appear on more than one section.
$\lbrace \text{A}, \text{B}, \text{C} \rbrace$
Find outcomes that are equally likely before you compare counts. The three letters are not, because they cover 5, 4 and 1 sections. The 10 sections are, as the problem states. So count sections, not letters.
Landing on A: 5 of the 10 equally likely sections show A and the other 5 do not. Exactly half of the sections give this event, so it is as likely as not.
$P(\text{land on A}) = \frac{1}{2}$
Landing on C: 1 of the 10 equally likely sections shows C. That is some, but far fewer than half, so the event is less likely to happen than not. It is unlikely.
$0 < P(\text{land on C}) < \frac{1}{2}$
Landing on D: D is not in the sample space, so no outcome fits this event. It is impossible.
$P(\text{land on D}) = 0$
Landing on A, B or C: every outcome in the sample space fits this event, so every spin makes it happen. It is certain.
$P(\text{land on A, B or C}) = 1$

Answer

The sample space is $\lbrace \text{A}, \text{B}, \text{C} \rbrace$ . Landing on A is as likely as not, with probability $\frac{1}{2}$ . Landing on C is unlikely. Landing on D is impossible, with probability $0$ . Landing on A, B or C is certain, with probability $1$ .

A probability places an event on a scale from $0$ to $1$ : $0$ for impossible, $\frac{1}{2}$ for as likely as not, and $1$ for certain. A sample space lists every possible outcome of one action, each written once. The list tells you what can happen. Comparing counts of outcomes tells you how likely an event is only when those outcomes are equally likely.

Next, you will find the exact probability of a simple event, and the probability that an event does not happen.