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6. Introduction to Probability
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6.04 Review: Probability concepts

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Experimental vs theoretical probability

We have learned about experimental probability and about theoretical probability and what the difference is between them.  The theoretical probability is what you expect to happen, but it isn't always what actually happens. What actually happens is called experimental probability.

For example, the table below shows the results after Kenya tossed the coin 20 times.  From our knowledge of theoretical probability, we expect that when we toss a fair coin the probability of getting tails should be equal to $1/2$1/2 .  In this experiment, however, the experimental probability of getting tails was actually $7/20$7/20 .  

Outcomes Frequency
Heads 13
Tails 7
Total 20

In theory, if Kenya flipped the coin a very large number of times we anticipate that the experimental probability would become closer to matching the theoretical probability of $1/2$1/2.

 

Summary

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

You may also see the term relative frequency which is the same as the experimental probability.

Practice questions

Question 1

A retail store served $773$773 customers in October, and there were $44$44 complaints during that month.

Determine, as a percentage, the experimental probability that a customer submits a complaint.

  1. Round your answer to the nearest whole percent.

Question 2

An insurance company found that in the past year, of the $2558$2558 claims made, $1493$1493 of them were from drivers under the age of 25.

Give your answers to the following questions as percentages, rounded to the nearest whole percent.

  1. What is the experimental probability that a claim is filed by someone under the age of 25?

  2. What is the experimental probability that a claim is filed by someone 25 or older?

Question 3

The experimental probability that a commuter uses public transport is $50%$50%.

Out of $500$500 commuters, how many would you expect to use public transport?

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 a number greater 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.

 

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 4

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

    A

    Unlikely

    B

    Even chance

    C

    Likely

    D

    Certain

    E

Question 5

Look at this spinner:

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

  1. What is the most likely symbol to spin?

    Ball.

    A

    Star.

    B

    Apple.

    C

    Pig.

    D

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

    Impossible

    A

    Unlikely

    B

    Even chance

    C

    Likely

    D

    Certain

    E

 

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 example

Question 6

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".

 

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 example

Question 7

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​

 

Practice questions

Question 8

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

  1. Impossible

    A

    Unlikely

    B

    Likely

    C

    Certain

    D

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

    A

    An orange ball

    B

    A blue pig

    C

    A green star

    D

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?

 

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