Is the Game Fair? Checking with Theoretical Probability

0:00This spinner has three equal parts: one red and two blue.

0:05Spin it twice. Player A wins if the colours match. Player B wins if they are different.

0:12A game is fair when each player has the same probability of winning.

0:17A common mistake is to think two colours means fifty-fifty. Let us check.

0:23An outcome is one possible result. Theoretical probability is the number of favourable outcomes divided by the total, when all outcomes are equally likely.

0:35Make a table: first spin down the side, second spin across the top. Three times three gives nine outcomes.

0:44The spins are independent events: the first does not change the second. So all nine outcomes are equally likely, one ninth each.

0:54Player A wins with red, red. That is one outcome, so one ninth.

1:00Player A also wins with blue, blue. Count them: one, two, three, four. That is four ninths.

1:09So the probability of a match is one ninth plus four ninths, which is five ninths.

1:16Player B wins when the colours are different: one, two, three, four outcomes, so four ninths.

1:24Check: five ninths plus four ninths makes nine ninths, which is one.

1:30Five ninths is more than four ninths, so Player A is more likely to win. The game is not fair.

1:38Two colours did not mean fifty-fifty: blue covers two of the three parts.

1:45Remember: a game is fair only when each player has the same probability of winning. This one is not.