Skip to content

Spinners

Probability · level 1, lesson 4 of 6

A question to start

Here are two spinners. You flick the arrow, it whizzes round and round, slows down, and stops pointing at one colour.

spinner Aspinner B

You win a sticker if the arrow stops on blue. Which spinner do you want? And on spinner A, how much more often does it land on orange than on blue: twice as often? Three times? A hundred times?

The idea

What a spinner is

A spinner is a circle cut into parts, with an arrow that can spin round its middle. You give the arrow a good flick. It turns many times, slows down, and stops. Wherever its point stops, that part is the result.

A spinner is like a coin or a bag of blocks: you cannot know where it will stop until it stops. It is a maybe machine.

Where does it land more often?

On spinner A, the orange part is big and the blue part is small. Most people feel that the arrow will land on orange more often. They are right. Here is the reason.

The arrow does not care about colours. It just spins and stops. And a circle is the same all the way round: no place on it is special. So the arrow is just as likely to stop anywhere round the circle.

Now draw dashed lines to cut spinner A into 4 parts that are exactly the same size. Each of these parts is called a quarter of the circle.

1234 = 4 equal parts4 equal blocks

Each quarter is the same size, so the arrow is just as likely to stop in any one of them. That gives 4 ways it can land, all equally likely.

  • Blue covers 1 quarter. That is 1 way.
  • Orange covers 3 quarters. That is 3 ways.

So orange has 3 ways and blue has 1 way. The arrow lands on orange about 3 times as often as on blue.

Look at the bag next to the spinner. It has 3 orange blocks and 1 blue block. That bag and spinner A are the same game: orange has 3 ways, blue has 1 way. A spinner is just a round bag, where the size of each part tells you how many ways it has.

So on a spinner: the bigger the part, the more likely the arrow lands there.

Spinner B

Spinner B is cut into 2 parts of the same size. Each part is called a half. Orange covers 1 half and blue covers 1 half. Each colour has 1 way out of 2. They are equally likely.

So if you want blue, pick spinner B. On spinner A, blue gets only 1 part out of 4. On spinner B, blue gets 1 part out of 2, which is a much bigger share of the circle.

A fair spinner

A spinner is fair when every colour has the same chance. That happens when every colour covers the same amount of the circle.

  • For 2 players: cut the circle into 2 halves, one colour each.
  • For 3 players: cut it into 3 equal parts, called thirds, one colour each.
  • For 4 players: cut it into 4 quarters, one colour each.
6 slices: fair for 23 thirds: fair for 3

The spinner on the left surprises some people. The colours are cut up into little slices. But orange has 3 slices and blue has 3 slices, all the same size. So orange covers just as much as blue, and the spinner is fair. It does not matter whether a colour is in one big piece or many small pieces. Only the total amount of the circle counts.

Being careful: equal sizes

To count slices, the slices must be the same size. Look at this spinner.

3 colours, but not 3 equal parts

It has 3 colours, so is each colour 1 way out of 3? No. The parts are not the same size. Orange is half the circle; blue and green are a quarter each. Cut the orange half in two with a line, and you have 4 equal quarters: orange 2, blue 1, green 1. Orange lands about twice as often as blue, and about twice as often as green. Blue and green are equally likely to each other.

So the rule is: cut the spinner into parts that are all the same size, then count.

Make your own spinner

You need a paper plate or a circle of card, a pencil and a paper clip.

  1. Fold the circle in half, then in half again. Open it. The folds cross at the middle and cut the circle into 4 equal quarters. (Folding makes the parts exactly equal, which is why it works so well.)
  2. Colour the quarters. Two orange and two blue makes a fair spinner for two players.
  3. Put the pencil point through one end of the paper clip, right on the middle of the circle. Hold the pencil still, standing up.
  4. Flick the paper clip. It spins round the pencil and stops pointing at a colour.

Things that can spoil a spinner: a pencil that is not in the very middle, a table that tilts, or a flick that is too gentle to go all the way round. A good spinner needs a strong flick every time.

Worked examples

Example 1. Which colour, how often?

A spinner is cut into 8 equal slices. 6 slices are orange and 2 are blue.

  1. The slices are equal, so each slice is 1 way. There are 8 ways.
  2. Orange has 6 ways. Blue has 2 ways.
  3. Orange is more likely. 6 is 3 times as much as 2, so orange lands about 3 times as often as blue.
  4. Notice: this is the same as spinner A. 6 out of 8 slices covers the same amount as 3 out of 4 quarters. Two spinners can look different and still be the same game.

Example 2. Make it fair.

A spinner has 4 equal quarters: 3 orange and 1 blue. How can you recolour it to make it fair for orange and blue?

  1. Fair means the same amount of each colour.
  2. There are 4 quarters, so each colour needs 2.
  3. Recolour one orange quarter blue. Now there are 2 orange and 2 blue. Fair.

Example 3. Three players.

Ali, Bea and Cy want a fair spinner. Ali suggests one with 6 equal slices: 2 for Ali, 2 for Bea, 2 for Cy. Is that fair?

  1. 6 equal slices, so 6 ways.
  2. Each person has 2 slices, so 2 ways each.
  3. Same number of ways for everybody. Fair. (It covers the same amount as one third each.)

Try it

  1. A spinner is half green and half orange. Which colour is more likely?
  2. A spinner has 4 equal quarters: 1 orange, 3 green. Which colour is more likely? About how many times as often?
  3. A spinner has 6 equal slices: 5 blue and 1 orange. You win on orange. Is it a good spinner for you?
  4. A spinner has 6 equal slices coloured orange, blue, orange, blue, orange, blue. Is it fair for orange and blue?
  5. A spinner is cut into 3 parts: orange is half the circle, and blue and green share the other half equally. Is it fair for 3 players?
  6. Draw a fair spinner for 4 players.
  7. A spinner has 8 equal slices: 4 orange, 2 blue and 2 green. Put the colours in order from most likely to least likely.
  8. On a fair spinner for orange and blue, can blue come up 3 times in a row?
Answers
  1. Neither. They each cover half the circle, so they are equally likely.
  2. Green: 3 ways against 1. Green lands about 3 times as often as orange.
  3. No. You have 1 way out of 6. Your friend (blue) has 5 ways. Blue is much more likely.
  4. Yes. Orange has 3 equal slices and blue has 3. It does not matter that they are split up.
  5. No. Orange covers half; blue and green each cover a quarter. The orange player wins about twice as often as each of the others. For 3 players you need 3 equal parts (thirds).
  6. Cut the circle into 4 equal quarters (fold it in half and in half again) and give each player a different colour, one quarter each.
  7. Orange (4 slices) is most likely. Blue and green (2 slices each) are equally likely, and less likely than orange.
  8. Yes. Each spin is a maybe, and blue is a maybe every time. Three blues in a row happens quite often. Fair means the same chance each spin, not taking turns. (More in the next lesson.)

Common mistake

Counting colours instead of space. A spinner with 3 colours is not always "1 way out of 3" for each. If the parts are different sizes, the bigger part wins more often. Always cut the spinner into equal parts first, then count.

Thinking a split-up colour is unfair. Three thin orange slices cover the same amount as one big orange slice of the same total size. The arrow does not care whether a colour is in one piece or many.

Where it leads