Outcomes and events

Chance in everyday words

People talk about chance all the time.

• It will probably rain this evening.
• Our team has a good chance today.
• I am sure the shop is open.

Words like probably, good chance and sure all measure how likely something is. Mathematics measures it with a number. Before we can find that number, we need a few words.

Experiments and outcomes

An experiment is an action whose result you cannot know before you do it. Rolling a die, spinning a spinner and picking a ticket from a box without looking are all experiments.

Each possible result is an outcome. The list of all the outcomes is often called the sample space.

• Rolling a die has six outcomes: $1, 2, 3, 4, 5, 6$.
• A spinner with ten equal sectors numbered $1$ to $10$ has ten outcomes.
• Picking one letter at random from the word RAIN has four outcomes: R, A, I and N.

Write the outcomes as a list, and check that you have them all. A missing outcome spoils every answer that follows.

Equally likely outcomes

A die is fair when it is a perfect cube of even material. Then no face has a better chance than another. We say the six outcomes are equally likely.

Many experiments give equally likely outcomes:

• spinning a spinner whose sectors are all the same size;
• picking one ticket from a box of identical tickets, without looking;
• choosing one student from a class "at random", which means every student has the same chance.

Other experiments do not give equally likely outcomes. Look at the two spinners in the picture.

The left spinner has eight equal sectors, so each number is equally likely. The right spinner has three outcomes, red, blue and yellow, but red covers half the circle. Red is more likely than blue. Its chance is not $\dfrac{1}{3}$.

Here are more outcomes that are not equally likely:

• whether it rains tomorrow or not;
• whether a drawing pin lands point up or point down;
• whether a cricket match ends in a win, a loss or a draw.

Listing two or three possible results does not make them equally likely. Ask whether anything makes one result easier than another.

Events

An event is a set of outcomes, described in words. For the die:

• the event "an even number" is made of the outcomes $2, 4, 6$;
• the event "a number more than $4$" is made of $5$ and $6$;
• the event "a $3$" is made of the single outcome $3$.

The outcomes that make an event happen are its favourable outcomes. An event with just one favourable outcome, such as "a $3$", is called an elementary event.

A worked example. A spinner has twelve equal sectors numbered $1$ to $12$. List the favourable outcomes for each event.

1. A multiple of $4$. $4, 8, 12$: three outcomes.
2. A number less than $5$. $1, 2, 3, 4$: four outcomes.
3. A prime number. $2, 3, 5, 7, 11$: five outcomes. Remember that $1$ is not prime.

Practise with new numbers

Try it in Python

A program can list the outcomes and pick out the favourable ones for you. This one uses a spinner numbered $1$ to $8$.

Common mistakes

• Missing an outcome. Picking a letter from APPLE has five outcomes, one for each letter, even though two of them are P.
• Assuming outcomes are equally likely. Win, lose and draw are three results, but they need not have the same chance.
• Counting $1$ as a prime number. It is not.
• Mixing up an outcome and an event. "An odd number" is an event made of three outcomes, $1$, $3$ and $5$.

Remember these

• An experiment has results you cannot know in advance. Each result is an outcome.
• Equally likely outcomes have the same chance: a fair die, equal sectors, identical tickets.
• An event is a set of outcomes. Its favourable outcomes are the ones that make it happen.
• An elementary event has exactly one favourable outcome.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. List the outcomes of picking one letter at random from the word TOWEL.

2. A die is rolled. Which outcomes favour the event "a number less than $3$"?

3. Are the outcomes "the bus is on time" and "the bus is late" equally likely?

4. A spinner has sectors of $200^\circ$, $100^\circ$ and $60^\circ$. Are its three colours equally likely?

5. Is "a $6$" on a die an elementary event?