Fair, or not fair?

The idea

Outcomes are equally likely when each has the same possibility of occurring; a fair (unbiased) coin or die, or a draw made at random, gives equally likely outcomes, but outcomes are not always equally likely.

Four red balls and one blue ball

A bag holds 4 red balls and 1 blue ball, all of the same size. You put your hand in and take out one ball without looking.

You will get a red ball or a blue ball.

Are "a red ball" and "a blue ball" equally likely?

• Yes, there are two colours, so each has the same chance
• No, a red ball is more likely
• No, a blue ball is more likely

Make red and blue equally likely

Use the − and + buttons to change the number of red and blue balls. Draw balls (each ball is put back) and look at the count. Your task: change the bag so that red and blue are equally likely.

Same number of balls, same chance

Outcomes are equally likely when each has the same chance of occurring. A fair coin, a fair die, or a ball drawn at random from a bag give equally likely outcomes. But outcomes are not always equally likely.

In the bag of 4 red and 1 blue, the two colours are not equally likely, because there are more red balls. The five balls, however, are equally likely: you draw without looking, so each ball has the same chance, 1 out of 5. Red and blue became equally likely only when there were as many red balls as blue balls.

A fair (unbiased) coin falls head or tail: 2 equally likely outcomes. A fair die shows 1, 2, 3, 4, 5 or 6: 6 equally likely outcomes. In the rest of this chapter we assume that all the outcomes of an experiment are equally likely.

Notes

Outcomes are equally likely when each has the same chance of occurring (a fair coin or die, a draw at random). Outcomes are not always equally likely: in a bag of 4 red and 1 blue ball, red and blue are not.

Check yourself

Which experiment has equally likely outcomes?

A bag has 4 red balls and no blue ball. How many blue balls must be put in so that red and blue become equally likely?

Answer: 4

Equally likely means the same number of balls of each colour: 4 red and 4 blue.

In a class of 40 students there are 25 girls and 15 boys. The teacher writes each name on an identical card, stirs the cards in a bag and draws one card. Which statement is true?

A fair coin is tossed 10 times and shows head 7 times. What does this tell us about the coin?

• Throwing a fair die once: the number on top — correct. Yes! Each of the numbers 1, 2, 3, 4, 5, 6 has the same possibility of showing up.
• Drawing a marble at random from a box of 3 blue, 2 white and 4 red marbles: its colour. No. There are 3 blue, 2 white and 4 red marbles, so the three colours are not equally likely. (Each of the 9 marbles is equally likely, but the colours are not.)
• Drawing a ball at random from a bag of 4 red and 1 blue ball: its colour. No. Red has 4 balls and blue has 1, so red is more likely than blue.
• "A girl" and "a boy" are equally likely, because there are two choices. No. There are 25 girl cards and only 15 boy cards, so a girl is more likely than a boy.
• Each of the 40 names is equally likely, and so are "a girl" and "a boy". The names are equally likely, but "a girl" has 25 names and "a boy" only 15, so these two events are not equally likely.
• Each of the 40 names is equally likely, but "a girl" and "a boy" are not — correct. Yes! The cards are identical and stirred, so each name has the same chance, 1 out of 40. But 25 of the names are girls and 15 are boys, so "a girl" is more likely.
• The coin is not fair, because head and tail are not equally likely. No. With only a few tosses the counts often differ from 5 and 5. A fair coin still gives head and tail the same chance on every toss.
• Nothing yet: a fair coin gives head and tail the same chance, even if a few tosses show 7 and 3 — correct. Yes! Equally likely is about the chance of each toss, not about getting exactly 5 and 5 in 10 tosses.
• The next toss must be a tail to make up for it. No. The coin does not remember. Each toss has the same chance of head and tail.