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8.01 Theoretical probability


The language of probability

Probability is the study of chance and prediction. To make sure our predictions are valid, we need to use the right mathematical language.

In general we will be thinking about a single test, known as a trial (also known as an experiment), that has more than one possible result, known as an outcome. A good example is flipping a coin:

Heads Tails

We say that flipping the coin is a trial, and there are two equally likely outcomes: head, and tails. The list of all possible outcomes of a trial is called the sample space.

Another example of a trial is rolling a die:

A single die All possible faces

There are $6$6 equally likely outcomes in the sample space$1$1$2$2$3$3$4$4$5$5, and $6$6. We can group these outcomes into events, such as "rolling an even number" or "rolling more than $3$3". Each outcome on its own is always an event, and sometimes events don't correspond to any outcomes.

We can think about different kinds of events that we care about, and sort them into categories of likelihood. Here are some examples when rolling a die:

Likelihood Event
Impossible Rolling a $9$9
Unlikely Rolling a $1$1
Even chance Rolling $4$4 or more
Likely Rolling a $2$2 or more
Certain Rolling between $1$1 and $6$6

What makes an event likely or unlikely depends on what happens when you repeat the trial many times. If the event happens more than half the time, we say it is likely, and if it happens less than half the time, we say it is unlikely. If it happens exactly half the time we say it has an even chance.

If every outcome corresponds to the event, we say it is certain. If no outcomes correspond to the event, we say it is impossible.



This is a full set of $52$52 playing cards:

Notice that there are many different events, depending on what result we are interested in:

  • Two colors
    • Red
    • Black
  • Four suits
    • Spades  
    • Hearts 
    • Clubs 
    • Diamonds 
  • Thirteen card values
    • The three "face cards"
      • "K" for "King" 
      • "Q" for "Queen" 
      • "J" for "Jack"
    • The numbers $2$2 through $10$10
    • "A" for "Ace", which is usually given the value of $1$1.

The deck of cards is shuffled, and the trial is going to be drawing a single card from the deck.

Here are some events sorted into each of the five likelihood categories:

Likelihood Event
Impossible Drawing a $17$17 of Hearts
Drawing a blue card
Drawing a $2$2 of Cups
Unlikely Drawing an Ace
Drawing a "face card"
Drawing a Spade
Even chance Drawing a black card
Drawing a red card
Likely Drawing a card numbered $2$2 through $10$10
Drawing a card that is not a $2$2
Drawing a card of any suit that is not Hearts
Certain Drawing a card that is a Spade, Heart, Club, or Diamond
Drawing a card that is either red or black

Drawing a "face card" is unlikely because there are fewer of them than the other cards. Drawing a black card has an even chance because there are just as many black cards as red cards. There are many more possible events we could describe, and fitting them into the right likelihood category can take some practice. We will investigate ways we can be precise in the next chapter.

Key words

Trial - a single experiment with different possible results. 

Outcome - the possible results of a trial.

Event - a grouping of outcomes. Each possible outcome is always an event on its own.

Likelihood - an event can be: 

  • impossible (can never happen) 
  • unlikely (happens less than half the time) 
  • even chance (happens half the time) 
  • likely (happens more than half the time) 
  • certain (always happens)

Sample space - a list of all the possible outcomes of a trial.

From, to, and between

Sometimes the language we use to describe chance can be less precise than we need it to be.

When we say "from $2$2 to $5$5" we mean including $2$2 and $5$5.

When we say "between $2$2 and $5$5 inclusive" we also mean including $2$2 and $5$5.

But when we say "between $2$2 and $5$5 exclusive" we mean numbers strictly greater than $2$2 and strictly less than $5$5 - that is, only the numbers $3$3 and $4$4.

We will not say "between $2$2 and $5$5" on its own because it isn't clear whether we include the ends or not.

Practice questions

Question 1

A six-sided die is rolled in a trial. What are the chances that the outcome is $2$2 or more?

A six-sided die with three sides shown. The numbers on these sides are 1, 3, and 5.
A three-dimensional illustration of a standard six-sided die. The die is primarily blue with white dots indicating the numbers on each visible face. Three faces are visible in the illustration showing 1 dot, 3 dots and five dots
  1. Impossible




    Even chance






Question 2

Look at this spinner:

A spinner whose possible outcomes are 3 balls, 2 pigs, 2 stars, 1 apple.

  1. Which symbol has the greatest chance to be spun?








  2. What is the likelihood of spinning a Ball.?





    Even chance







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.

Outcomes and events

In the section above we looked at the difference between an outcome and an event.

An outcome represents a possible result of a trial. When you roll a six-sided die, the outcomes are the numbers from $1$1 to $6$6.

An event is a grouping of outcomes. When you roll a six-sided die, events might include "rolling an even number", or "rolling more than $5$5".

Each outcome is always an event - for example, "rolling a $6$6" is an event.
But other events might not match the outcomes at all, such as "rolling more than $6$6".


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

Question 3

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".

Question 4

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 "favorable outcomes" and the rest. Then we can use the formula:

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

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


Worked examples

Question 5

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

Question 6

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 "favorable outcomes", and there are $3+3=6$3+3=6 all together.

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

Question 7

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 "favorable 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 8

A probability of $\frac{4}{5}$45 means the event is:

  1. Impossible








Question 9

Select the two events which have a probability of $25%$25% on this spinner:

A colorful prize wheel with three distinct sections, each a different color and featuring a unique symbol. The largest section is red with a orange ball symbol, occupying about half of the wheel. The yellow section features a red apple, and the blue section has a blue pig symbol; both of these sections are smaller and appear to be equal in size, each making up a quarter of the wheel.

  1. A red apple


    An orange ball


    A blue pig


    A green star


Question 10

A jar contains $10$10 marbles in total. Some of the marbles are blue and the rest are red.

  1. If the probability of picking a red marble is $\frac{4}{10}$410, how many red marbles are there in the jar?

  2. What is the probability of picking a blue marble?



Theoretical probability

We can make predictions for trials by first creating the sample space and then determining the theoretical probability of each outcome.


If you roll two six-sided dice and add the numbers together, what is the probability of getting a sum of $6$6? What about a sum of $10$10 or greater? $8$8 or less?

Before we answer these questions we need to determine the sample space. The possible outcomes for two dice can be drawn in a grid:

    Second die
    $1$1 $2$2 $3$3 $4$4 $5$5 $6$6
First die $1$1 $1,1$1,1 $1,2$1,2 $1,3$1,3 $1,4$1,4 $1,5$1,5 $1,6$1,6
$2$2 $2,1$2,1 $2,2$2,2 $2,3$2,3 $2,4$2,4 $2,5$2,5 $2,6$2,6
$3$3 $3,1$3,1 $3,2$3,2 $3,3$3,3 $3,4$3,4 $3,5$3,5 $3,6$3,6
$4$4 $4,1$4,1 $4,2$4,2 $4,3$4,3 $4,4$4,4 $4,5$4,5 $4,6$4,6
$5$5 $5,1$5,1 $5,2$5,2 $5,3$5,3 $5,4$5,4 $5,5$5,5 $5,6$5,6
$6$6 $6,1$6,1 $6,2$6,2 $6,3$6,3 $6,4$6,4 $6,5$6,5 $6,6$6,6

We can now tell that there are $36$36 possible outcomes. Depending on the trial we can highlight the favorable outcomes corresponding to the event, and the probability of any particular event is given by the formula

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{36}$Probability=Number of favorable outcomes36

Explore this applet to find the various probabilities:

Once we have a sample space with every outcome being equally likely, we can express the probability as a fraction, decimal, or percentage.


Practice questions

Question 11

Paul has $23$23 marbles in a bag. $4$4 of them are black. Paul picks a marble from the bag without looking.

  1. What is the probability that Paul picks a black marble?

Question 12

An eight-sided die is rolled $25$25 times. How many times should we expect to roll a $7$7? Round your answer to the nearest whole number.

 an eight-sided die
eight-sided die

Question 13

The following twelve-sided die is rolled.

A numbered twelve-sided die.
  1. What is the chance of rolling five or more?

  2. What is the chance of rolling less than five?



Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

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