Math Core

Will it rain tomorrow? Will your team win? Will the spinner land on blue? Probability is a number that tells you how likely something is to happen, so you can compare chances instead of just guessing.

The likelihood scale

A probability is always a number from $0$ to $1$ . You can write it as a fraction, a decimal or a percent.

0 means impossible, 1/2 means equally likely to happen or not, 1 means certain

Probability	Description	Example
$0$	impossible	rolling a $7$ on a standard number cube
close to $0$	unlikely	picking the one red marble from a bag of $50$
$\dfrac{1}{2}$	equally likely as not	a fair coin landing heads
close to $1$	likely	picking a blue marble from a bag of $49$ blue and $1$ red
$1$	certain	rolling a number less than $7$ on a standard number cube

The closer a probability is to $1$ , the more likely the event. The closer it is to $0$ , the less likely.

Outcomes, sample spaces and events

To find a probability, you first need words for what can happen.

Definition

Sample space

An outcome is one possible result of a chance experiment. The sample space is the list of all possible outcomes. An event is a set of outcomes you care about.

When you roll a standard number cube, the sample space is $\{1, 2, 3, 4, 5, 6\}$ . The event "roll an even number" is made of the outcomes $2$ , $4$ and $6$ . The outcomes in an event are sometimes called favorable outcomes.

Finding a probability

When every outcome in the sample space has the same chance (a fair coin, a fair number cube, a spinner with equal sections, a marble picked without looking), you can count.

Probability of an event

If all outcomes are equally likely,

P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}.

The symbol $P(\text{even})$ is read "the probability of rolling an even number."

Worked example: Rolling a number cube

You roll a standard number cube. Find $P(\text{even})$ and $P(\text{greater than } 4)$ .

The sample space has $6$ outcomes.

Even numbers: $2, 4, 6$ . That is $3$ favorable outcomes, so $P(\text{even}) = \dfrac{3}{6} = \dfrac{1}{2}$ .
Greater than $4$ : $5, 6$ . That is $2$ favorable outcomes, so $P(\text{greater than } 4) = \dfrac{2}{6} = \dfrac{1}{3}$ .

An even number is more likely, because $\dfrac{1}{2} > \dfrac{1}{3}$ .

Worked example: Picking a marble

A bag holds $3$ red, $5$ green and $2$ white marbles. You pick one without looking. What is the probability that it is green? Write your answer as a fraction, a decimal and a percent.

There are $3 + 5 + 2 = 10$ marbles, and $5$ are green.

P(\text{green}) = \frac{5}{10} = \frac{1}{2} = 0.5 = 50\%.

Common mistake

Count every outcome in the denominator, not just the other colors. In the bag above, $P(\text{green})$ is $\dfrac{5}{10}$ , not $\dfrac{5}{5}$ (green compared with the non-green marbles). The bottom of the fraction is always the total.

The complement

Every outcome either is in an event or is not. The event "not $A$ " is called the complement of $A$ . Since one of the two must happen, their probabilities add to $1$ :

P(\text{not } A) = 1 - P(A).

Worked example: Using the complement

The probability that a bus is late is $0.15$ . What is the probability that it is not late?

P(\text{not late}) = 1 - 0.15 = 0.85.

So the bus is on time $85\%$ of the time, which is likely.

Tip

Check your answer: a probability can never be negative and can never be greater than $1$ . If you get $\dfrac{7}{5}$ or $-0.2$ , something went wrong.

Practice

Practice 1

The probability that Maya's soccer team wins its next game is $0.9$ . Which word best describes this event?

Practice 2

You roll one standard number cube. Which event is impossible?

Practice 3

You roll a standard number cube. What is the probability of rolling a prime number? Give your answer as a fraction.