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.

 

Exploration

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.

Practice questions

Question 1

Question 2

 

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

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

Question 9

Question 10

 

 

Theoretical probability

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

Exploration

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:

Created with Geogebra

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

Question 12

Question 13