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Rolling a die

Probability · level 1, lesson 6 of 6

A question to start

Here are two games. Both use one ordinary die.

  • Game 1: roll the die. If it shows 6, you win a point.
  • Game 2: roll the die. If it shows an even number, you win a point.

You play each game 12 times. In which game will you win more points? About how many more?

Many board games make you wait for a 6 before you can start, and the wait can feel like forever. Is 6 harder to roll than the other numbers? Or does it just feel that way?

The idea

What a die is

A die is a small cube with dots on its faces. (Two of them are called dice. Many people say "a dice" for one, and everyone understands, but "die" is the old word for one.) The six faces show 1, 2, 3, 4, 5 and 6 dots.

You shake the die in your hand and throw it so that it tumbles and rolls. When it stops, the face on top is the result.

Why each face is just as likely

Look at the shape of a die.

A die is a cube. Every face of a cube is the same size: a square. Every edge is the same length. Every corner is the same. If you painted out the dots, you could not tell one face from another. You could turn the cube any way you like and it would look exactly the same.

So when a die tumbles, no face is special. Nothing pushes it to land on 6 more than on 1. The dots do not matter to the tumbling; they are only there so we can read the result. That is why we say each face is equally likely: each of the 6 faces has the same chance of ending up on top.

This is the same kind of reason as for a spinner (every part of the circle is the same) and a bag of blocks (every block feels the same). When things are the same in every way except the label, they are equally likely.

A die only works this way if it really is a fair cube. A die that is chipped, bent, or made heavier on one side (people who cheat have done this) can land on some faces more often. Real dice from a shop are made very carefully so that they are fair.

A die fact: opposite faces

Here is something to check on a die at home. Look at the face with 1 dot, and turn the die over to see the face on the opposite side. It has 6 dots. Now look at 2: opposite is 5. And 3: opposite is 4.

1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7. On an ordinary die, opposite faces always add to 7. Dice makers do this on purpose, so that the big numbers and small numbers are spread all round the cube.

Game 1: "6 wins"

In game 1, you win only on a 6.

Count the ways: 6 faces, all equally likely. Only 1 of them wins. So you win in 1 way out of 6.

That means that, on average, you win about once every 6 rolls. In 12 rolls, that is about 2 wins: once in the first 6 rolls and once in the next 6, roughly speaking.

Game 2: "even wins"

In game 2, you win on an even number. Even numbers are the ones you can share into two equal groups: 2, 4 and 6. (Odd numbers, 1, 3 and 5, always have one left over.)

Count the ways: 3 faces win (2, 4 and 6) out of 6. So you win in 3 ways out of 6. That is half the faces, so you win about half the time. In 12 rolls, that is about 6 wins.

Which game wins more often?

Game 2 wins in 3 ways. Game 1 wins in 1 way. So game 2 wins about 3 times as often. In 12 rolls, about 6 points in game 2 against about 2 in game 1.

Look at the pictures again. The 6 is coloured in both of them. Game 2 includes everything game 1 has, and more: it wins on 6 too, and also on 2 and 4. So game 2 can never be the worse game. Every roll that wins game 1 also wins game 2.

Is 6 hard to roll?

Now back to the board game question. Is 6 harder to roll than other numbers? No. A 6 is exactly as likely as a 1, or a 3, or any other face. The 6 only feels hard because you are waiting for one particular face out of six. If the game said "wait for a 1", the wait would be just as long.

And the wait can be long. "1 way out of 6" does not mean you get a 6 in every 6 rolls. Sometimes it comes on the first roll. Sometimes you wait 10 rolls or more. If you roll 6 times, there is about a 1 in 3 chance of getting no 6 at all. Even in 12 rolls, about 1 time in 9 there is no 6. It is not that the die is against you. That is just how a maybe works.

Worked examples

Example 1. Count the ways.

You roll a die. How many ways are there to roll a number smaller than 3?

  1. List the faces smaller than 3: 1 and 2.
  2. That is 2 ways out of 6.
  3. Compare with "rolling a 6" (1 way). Smaller than 3 is twice as likely.

Example 2. Which is more likely?

Which is more likely: rolling an odd number, or rolling a number bigger than 4?

  1. Odd: 1, 3, 5. That is 3 ways.
  2. Bigger than 4: 5, 6. That is 2 ways.
  3. 3 is more than 2, so odd is more likely.

Example 3. Make a fair game.

Two players share one die. Make a fair game, where each has the same chance.

  1. Fair means the same number of ways for each player.
  2. There are 6 faces, so give each player 3.
  3. One fair game: player A wins on 1, 2, 3, and player B wins on 4, 5, 6.
  4. Another: A wins on odd (1, 3, 5), B wins on even (2, 4, 6).
  5. Not fair: A wins on 6, B wins on 1, 2, 3, 4, 5. That is 1 way against 5.

Try it

  1. How many faces does a die have? How many dots are there on the face opposite the 2?
  2. Is rolling a 1 more likely, less likely, or just as likely as rolling a 6?
  3. How many ways are there to roll an even number? An odd number?
  4. Which is more likely: rolling a 3, or rolling a number bigger than 3?
  5. You play "6 wins" for 18 rolls. About how many times do you expect to win?
  6. Jo says, "I rolled 6 times and got no 6. The die must be broken." Is she right?
  7. Make a game with one die where player A is twice as likely to win as player B, and every roll has a winner.
  8. A die is rolled once. Which is more likely: "a number from 1 to 6" or "a number smaller than 7"?
Answers
  1. A die has 6 faces. Opposite the 2 is the 5 (opposite faces add to 7, and 2 + 5 = 7).
  2. Just as likely. Each face has 1 way out of 6.
  3. Even: 2, 4, 6, so 3 ways. Odd: 1, 3, 5, so 3 ways. The same.
  4. Bigger than 3: that is 4, 5, 6, so 3 ways. Rolling a 3 is only 1 way.
  5. 1 win in every 6 rolls, roughly, and 18 is 6 + 6 + 6, so about 3 wins. It might be 1, 2, 4 or 5; about 3 is the best guess.
  6. No. Getting no 6 in 6 rolls happens about 1 time in 3 with a perfectly fair die. That is not strange at all.
  7. There are 6 faces, so give A 4 faces and B 2 faces: 4 ways is twice 2 ways. For example, A wins on 1, 2, 3, 4, and B wins on 5, 6.
  8. Neither: both are sure! Every face is from 1 to 6, and every face is smaller than 7. Both have all 6 ways.

Common mistake

Thinking 6 is harder to roll than other numbers. Every face of a fair die is equally likely. Waiting for a 6 feels long because you need one face out of six, and any single face would feel the same.

Thinking "1 out of 6" means "once in every 6 rolls". It means about once in 6 rolls on average. In any 6 rolls you might get no 6, or two, or even three.

Where it leads