Which coin is fair?

The idea

A fair or unbiased coin is symmetrical, so there is no reason for it to come down more often on one side than the other; a random toss lets it fall freely without bias or interference.

Two look-alike coins

Two coins look exactly alike. One is fair. The other has a hidden weight on one side. How could you find out which is fair?

What is a good way to tell the two coins apart?

• Toss each one many times and compare
• Toss each once
• Look at them closely

Test the coins

Toss coin A and coin B at least 200 times each. Then say which coin is fair.

No favourite side

A fair (unbiased) coin is symmetrical: there is no reason for it to land on one side more often than the other. A random toss lets the coin fall freely, with no interference.

A fair coin has P(heads) = P(tails) = 1/2, so neither side is favoured. A coin that keeps giving far more heads than tails is biased.

Notes

A fair (unbiased) coin is symmetrical, so heads and tails are equally likely. A random toss means the coin falls freely, with no interference.

Check yourself

A coin is tossed 100 times and shows 52 heads. Is it likely to be fair?

A coin is tossed 100 times and shows 90 heads. What would you suspect?

What does it mean to say a coin is fair?

For a fair coin, what is the probability of heads, as a decimal?

Answer: 0.5

There are 2 equally likely outcomes and 1 is heads, so P(heads) = 1/2 = 0.5.

• No. It should be exactly 50. Even a fair coin rarely gives exactly 50 heads in 100 tosses.
• Yes. 52 is close to 50 — correct. Yes. This is what we expect from a fair coin.
• No. A fair coin gives only heads. A fair coin gives heads and tails about equally often.
• The coin is fair. A fair coin would give close to 50 heads.
• The coin has no memory. It is true that coins have no memory, but that does not explain 90 heads.
• The coin is biased — correct. Yes. 90 heads in 100 tosses is far from what a fair coin does.
• It is symmetrical, so heads and tails are equally likely — correct. Yes.
• It shows heads and tails in turn. The results of a fair coin are random.
• It always lands on its edge. A fair coin falls on one of its two faces.