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Play: Fair or unfair?

Probability · step 1 of 5

A question to start

Let us play a game with one ordinary dice. I roll it. If it shows 1, 2, 3, 4 or 5, I win. If it shows 6, you win. Do you want to play?

You should say no! Five of the six faces make me win, and only one makes you win. Each face is just as likely to come up as any other, so I will win much more often than you. This game is unfair.

Thinking about how likely things are, and whether a game gives everyone the same chance, is the start of probability: the mathematics of chance.

The idea

Some things are sure, some are impossible, most are in between

Here are some words for how likely something is:

  • Impossible: it can never happen. Rolling a 7 on an ordinary dice.
  • Unlikely: it can happen, but usually does not. Rolling a 6.
  • Even chance: it is just as likely to happen as not. A coin landing heads.
  • Likely: it usually happens. Rolling a number smaller than 6.
  • Certain: it is sure to happen. Rolling a number smaller than 7.

You can put them on a chance line, from impossible at one end to certain at the other.

impossibleunlikelyeven chance likelycertain

Counting the ways

How do you know where something goes on the line? Count the ways it can happen, out of all the ways things could turn out, when every way is just as likely.

A dice has 6 faces. "Rolling a 6" can happen in 1 way out of 6. "Rolling an even number" can happen in 3 ways out of 6 (2, 4 or 6): that is half of them, an even chance.

The more ways something can happen, the more likely it is.

Fair games

A game is fair when every player has the same chance of winning. The game at the top is unfair: I have 5 ways to win and you have 1.

You can make it fair by sharing out the faces equally:

  • I win on 1, 2 or 3; you win on 4, 5 or 6. Three ways each. Fair.
  • I win on an odd number; you win on an even number. Three ways each. Fair.

Spinners and bags

A spinner is fair to each colour only if the colours cover equal parts of it. A spinner that is three quarters orange and one quarter blue lands on orange much more often.

fairnot fair

A bag of marbles works the same way. In a bag with 3 orange marbles and 1 blue marble, if you pull one out without looking, orange is likely and blue is unlikely, because there are 3 orange ways and only 1 blue way.

Worked example

Sam and Lee play a game with two coins. They toss both. If the coins match (both heads or both tails), Sam wins. If they are different, Lee wins. Is it fair?

  1. List every way the two coins can land. Call the coins the first coin and the second coin, so you do not miss anything:
    • first heads, second heads (match);
    • first heads, second tails (different);
    • first tails, second heads (different);
    • first tails, second tails (match).
  2. Each of these 4 ways is just as likely.
  3. Sam has 2 ways (the two matches). Lee has 2 ways (the two differences).
  4. Two ways each: the game is fair.

Many people think "there are three results: two heads, two tails, or one of each, so Lee has only 1 chance in 3". But "one of each" can happen in two ways (heads then tails, or tails then heads). Listing the coins separately shows this clearly.

Try it

  1. Where on the chance line do these go? (a) Rolling a 0 on a dice. (b) Picking an orange marble from a bag of 9 orange and 1 blue. (c) A coin landing tails.
  2. A bag has 2 orange marbles and 5 blue ones. Which colour are you more likely to pick?
  3. Is this game fair? Roll a dice. You win on 1 or 2; your friend wins on 3, 4, 5 or 6. If not, how could you change it?
  4. A spinner has 4 equal parts: 3 green and 1 blue. Green wins for one player, blue for the other. Is it fair?
  5. A coin has landed heads 4 times in a row. Is tails now more likely on the next toss?
Answers
  1. (a) Impossible: no face has 0 dots. (b) Likely: 9 ways out of 10. (c) Even chance.
  2. Blue: there are 5 blue ways and only 2 orange ways.
  3. Not fair: 2 ways against 4 ways. Make it fair by giving each player 3 faces, for example you win on 1, 2 or 3.
  4. Not fair: green has 3 parts and blue only 1. Colour the parts 2 green and 2 blue to make it fair.
  5. No. The coin does not remember what happened before. Every toss is still an even chance: 1 way for heads and 1 way for tails.

Common mistake

Thinking that luck "evens out" on the next try. After 4 heads in a row, many people feel tails is "due". But a coin has no memory: each toss starts fresh, with an even chance either way. Over a very large number of tosses, heads and tails do come out close to half each, but that is because the later tosses swamp the early ones, not because the coin makes up for them.

Where it leads