5.01 Review: Concepts of 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.

 

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

 

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.

 

Repeating trials

Once we know the probability of an event, we can predict how many times this event will occur if a trial is repeated several times.

We multiply the probability of the event by the number of trials, rounding to the nearest whole number.

Worked example

A letter is chosen at random from the word "MATHEMATICS" two hundred times. and the results written down in a list.

How many times can we expect that a "T" will be chosen? Round your answer to the nearest whole number.

Think: There are $11$11 letters in the word "MATHEMATICS", and each one is equally likely to be chosen. We can think of the sample space like this spinner:

Once we know the probability of "T" in one trial, we can multiply it by $200$200 to find the expected number of times that "T" will appear in the list.

Do: Using the formula,

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{2}{11}$Probability=Number of favorable outcomesTotal number of outcomes​=211​

This means each time a letter is chosen, there is a $\frac{2}{11}$211​ chance of choosing a "T".

Multiplying this probability by the number of trials ($200$200) tells us the expected number of times that "T" will be chosen:

$\text{Number of Ts}=200\times\frac{2}{11}=\frac{400}{11}$Number of Ts=200×211​=40011​

As a decimal this is $36.\overline{36}$36.36, which rounds to the nearest whole number $36$36. This is how many times we should expect "T" to be chosen in the list of $200$200 letters.

 

Practice questions

Question 10

Question 11

 

Experimental probability

In order to predict the future, we sometimes need to determine the probability by running experiments, or looking at data that has already been collected. This is called experimental probability, since we determine the probability of each outcome by looking at past events.

Exploration

Imagine we have a "loaded" die, where a weight is placed inside the die opposite the face that the cheater wants to come up the most (in this case, the $6$6):

If the die is made like this, the probability of each outcome is no longer equal, and we cannot say that the probability of rolling any particular face is $\frac{1}{6}$16​.

Instead we will need to roll the die many times and record our results, and use these results to predict the future. Here are the results of an experiment where the die was rolled $200$200 times:

Result	Number of rolls
$1$1	$11$11
$2$2	$19$19
$3$3	$18$18
$4$4	$18$18
$5$5	$20$20
$6$6	$114$114

We can now try to predict the future using this experimental data, and the following formula:

$\text{Experimental probability of event}=\frac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}$Experimental probability of event=Number of times event occurred in experimentsTotal number of experiments​

Here is the table again, with the experimental probability of each face listed as a percentage:

Result	Number of rolls	Experimental probability
$1$1	$11$11	$5.5%$5.5%
$2$2	$19$19	$9.5%$9.5%
$3$3	$18$18	$9%$9%
$4$4	$18$18	$9%$9%
$5$5	$20$20	$10%$10%
$6$6	$114$114	$57%$57%

A normal die has around $17%$17% chance of rolling a $6$6, but this die rolls a $6$6 more than half the time!

Sometimes our "experiments" involve looking at historical data instead. For example, we can't run hundreds of Eurovision Song Contests to test out who would win, so instead we look at past performance when trying to predict the future. The following table shows the winner of the Eurovision Song Contest from 1999 to 2018:

Year	Winning country	Year	Winning country
1999	Sweden	2009	Norway
2000	Denmark	2010	Germany
2001	Estonia	2011	Azerbaijan
2002	Latvia	2012	Sweden
2003	Turkey	2013	Denmark
2004	Ukraine	2014	Austria
2005	Greece	2015	Sweden
2006	Finland	2016	Ukraine
2007	Serbia	2017	Portugal
2008	Russia	2018	Israel

What is the experimental probability that Sweden will win the next Eurovision Song Contest?

We think of each contest as an "experiment", and there are $20$20 in total. The winning country is the event, and we can tell that $3$3 of the contests were won by Sweden. So using the same formula as above,

$\text{Experimental probability of event}=\frac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}$Experimental probability of event=Number of times event occurred in experimentsTotal number of experiments​

the experimental probability is $\frac{3}{20}$320​, which is $15%$15%.

How many of the next $50$50 contests can Sweden expect to win?

Just like in the last chapter, we can calculate this by multiplying the experimental probability of an event by the number of trials. In this case Sweden can expect to win

$\frac{3}{20}\times50=\frac{150}{20}$320​×50=15020​ contests

This rounds to $8$8 contests out of the next $50$50.

Practice questions

Question 12

Question 13