5.01 Introduction to 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 that has more than one possible result, known as an outcome. A trial is a repeated part of an experiment that is repeated over and over. A good example is flipping a coin:

We say that flipping the coin is a trial, and there are two equally likely outcomes: heads 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:

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

We can think about different kinds of events that we care about, and sort them into categories of likelihood. The likelihood of an event depends on what happens when you repeat the trial many times.

• Impossible: no outcomes correspond to the event

• Unlikely: the event happens less than half the time

• Equally likely: the event happens the same number of times as the other events

• Likely: the event happens more than half the time

• Certain: every outcome corresponds to the event

Here are some examples when rolling a die:

Examples

Example 1

A six-sided die is rolled in a trial. Describe the likelihood that the outcome is 2 or more.

Worked Solution

Create a strategy

First, we want to list the sample space (all the possible outcomes) of rolling a die. Then, we want to count the number of outcomes that are 2 or more and compare it to the total number of outcomes.

• Impossible: no outcomes correspond to the event

• Unlikely: the event happens less than half the time

• Equally likely: the event happens the same number of times as the other events

• Likely: the event happens more than half the time

• Certain: every outcome corresponds to the event

Apply the idea

The sample space of rolling a die is \{1,2,3,4,5,6\}. This shows there are 6 possible outcomes.

There are 5 outcomes that result in 2 or a number greater than 2, which is more than half of the outcomes.

The chances of rolling a 2 or more is likely.

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Example 2

Look at this spinner:

a

Which symbol is most likely to be spun?

Worked Solution

Create a strategy

This question is asking us to find the symbol that has more outcomes than the other symbols on the spinner. This will be the symbol that appears most on the spinner.

Apply the idea

The spinner has 8 sectors that show a ball, an apple, a pig, or a star.

• There are 3 sections or outcomes with a ball

• There is 1 section or outcome with an apple

• There are 2 sections or outcomes with a pig

• There are 2 sections or outcomes with a star

The ball has more outcomes than the other symbols, so the most likely symbol to spin is a ball.

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b

Describe the likelihood of spinning a ball.

Worked Solution

Create a strategy

In the previous part, we found that the sample space has 8 outcomes and spinning a ball has 3 outcomes.

Likelihood can be described as impossible, unlikely, equally likely, likely, or certian.

Apply the idea

Spinning a ball has only 3 outcomes, which is less than half of the total number of outcomes \left(8\right).

So, the likelihood of spinning a ball is unlikely.

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