23.03 Probability as a number

The likelihood of an event after a trial can be placed on a spectrum from $0$0 to $1$1 using fractions or decimals, or from $0%$0% to $100%$100% using percentages:

A probability can never be less than $0$0 or more than $1$1. The larger the number, the more likely it is, and the smaller the number, the less likely it is. We will now look at how to determine these numbers exactly.

 

Equal probabilities

If every outcome in a trial is equally likely, then the probability of one particular outcome is given by the equation:

$\text{Probability}=\frac{1}{\text{Size of sample space}}$Probability=1Size of sample space​

Remember that the sample space is the list of all possible outcomes. We can multiply this number by $100%$100% to find the probability as a percentage.

Worked examples

Example 1

What is the probability of rolling a $4$4 on a $6$6-sided die?

Think: There are $6$6 outcomes in the sample space: $1$1, $2$2, $3$3, $4$4, $5$5, $6$6. We will use the formula above.

Do: Probability $=$= $\frac{1}{6}$16​

Reflect: We will often say this kind of probability in words like this:

"There is a $1$1 in $6$6 chance of rolling a $4$4".

Example 2

What is the probability of spinning a Star  on this spinner?

Express your answer as a percentage.

Think: The list of events is:

, , , ,

The size of the sample space is $5$5, and each outcome is equally likely.

Do: Probability $=$= $\left(\frac{1}{5}\times100\right)%=20%$(15​×100)%=20%.

Reflect: We will often say this kind of probability in words like this:

"There is a $20%$20% chance of spinning a Star "

 

Unequal probabilities

If the outcomes in a sample space are not equally likely, then we have to think about splitting the sample space up into "favourable outcomes" and the rest. Then we can use the formula:

$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes​

If every outcome is favourable, then we have a probability of $1$1. If there are no favourable outcomes, the probability is $0$0.

Worked examples

Example 3

What is the probability of spinning a Pig  on this spinner?

Think: We can think about this spinner as having five possible events:

, , , ,

But we can tell that spinning a Pig is more likely than the other outcomes. It is more useful to think about the sample space instead, which has $6$6 sectors, and $2$2 of them have a Pig .

Do: Probability $=$= $\frac{2}{6}=\frac{1}{3}$26​=13​

Example 4

What is the probability of spinning a Star or an Apple on this spinner?

Express your answer as a decimal.

Think: There are $10$10 different sectors, $3$3 of them have a Star and $3$3 of them have an Apple . These are the "favourable outcomes", and there are $3+3=6$3+3=6 all together.

Do: Probability $=$= $\frac{6}{10}=0.6$610​=0.6

Example 5

What is the probability of drawing a card from a standard deck of $52$52 cards, that is red and has an even number on it?
What is the probability of drawing a Club that is not the Jack?

Which is more likely?

Think: For the first trial, the cards with even numbers are $2$2, $4$4, $6$6, $8$8, and $10$10. Each of these numbers appear $4$4 times, once for each suit, and $2$2 of the suits are red. This means there are $10$10 cards we could draw corresponding to the event we want, the "favourable outcomes".

For the second trial, there are $13$13 cards in each suit, and $12$12 of them are not the Jack.

Do: For the first trial, Probability $=$= $\frac{10}{52}$1052​.

For the second trial, Probability $=$= $\frac{12}{52}$1252​.

Since the second trial has a higher probability of success, it is more likely that we draw a Club that is not the Jack of Clubs. 

Reflect: We could simplify the two fractions to $\frac{5}{26}$526​ and $\frac{3}{13}$313​, but this makes it harder to compare probabilities. Often it is better to not simplify fractions in this topic.

Practice questions

Question 1

Question 2

Question 3