Experiments, outcomes and events

When you throw a die, you know what can come up – a number from 1 to 6 – but not which number it will be.

Throw the die a few times. Before each throw, try to guess what will come up.

An experiment is an action whose result is not known in advance. For example:

• throwing a die;
• tossing a coin – it lands heads or tails;
• drawing one pupil from the class;
• spinning a spinner: a circle split into parts, with an arrow at its centre that you spin. The result is the part the arrow stops on.

Each thing that can come up in an experiment is an outcome. All the outcomes of the experiment together are its sample space, written in curly brackets:

• throwing a die: $\displaystyle \{1,2,3,4,5,6\}$ – six outcomes;
• the spinner in the picture: $\displaystyle \{1,2,3,4\}$ – four outcomes;
• tossing a coin: two outcomes, heads and tails;
• drawing a pupil from a class of 30: 30 outcomes – each pupil is an outcome.

An event is some of the outcomes of the experiment. An event happens when the outcome that came up is one of its outcomes.

• When a die is thrown, the event “an even number comes up” is $\displaystyle \{2,4,6\}$.
• A 4 comes up – the event happened.
• A 3 comes up – the event did not happen.
• An event is usually called by a capital letter: $\displaystyle A$ – “an even number comes up”.

An event can hold all the outcomes of the experiment, or none of them:

• a certain event – all the outcomes. It always happens. With a die: “a number less than 7 comes up”.
• an impossible event – no outcome at all. It never happens. With a die: “an 8 comes up”.

The example below writes the sample space of drawing a card, and the outcomes of three events.

A box holds 10 cards, numbered 1 to 10.
One card is drawn from the box without looking.

• The sample space: $\displaystyle \{1,2,3,4,5,6,7,8,9,10\}$ – ten outcomes.
• “The number is divisible by 3”: $\displaystyle \{3,6,9\}$.
• “The number is greater than 7”: $\displaystyle \{8,9,10\}$.
• “The number is less than 1”: no outcome at all – an impossible event.

A die is thrown.
Write the outcomes of each event.

An odd number comes up.

A number greater than 4 comes up.

A number divisible by 3 comes up.

A box holds 20 cards, numbered 1 to 20.
One card is drawn without looking.

How many outcomes are in the sample space?

Write the outcomes of the event “the number is divisible by 5”.

Write the outcomes of the event “the number has two digits and is divisible by 4”.

A die is thrown.
For each event, write whether it is certain, impossible, or neither.

A number less than 10 comes up.

A 0 comes up.

A number less than 6 comes up.

A number that is both even and odd comes up.

A die is thrown, and a 4 comes up.
Which of these events happened?

• $\displaystyle A$: an even number comes up.
• $\displaystyle B$: a number greater than 4 comes up.
• $\displaystyle C$: a number less than 5 comes up.
• $\displaystyle D$: a 1 or a 4 comes up.

There are 32 pupils in Noa’s class.
One pupil is drawn from the class for classroom duty.

What is the sample space? How many outcomes does it hold?

Write a certain event and an impossible event of this experiment.