FILE No.125Unsolved
Player A and Player B roll two colored dice at the same time. If the colors match, A wins; if they differ, B wins.
One die has 5 red faces and 1 blue face.
How many blue faces should the other die have so that both players are equally likely to win (a fair game)?
If you get stuck, open the hints one at a time, in order.
How many combinations give different colors in total?
Let the number of blue faces be □ and write an equation.
The game is fair when the probability of different colors is 1/2.
How many blue faces should the other die have so that both players are equally likely to win (a fair game)?
Choose the answer closest to the truth.
3 faces
The answer is "3 faces."
Let the other die have □ blue faces, so it has (6 − □) red faces. The probability the colors differ is (5 × □ + 1 × (6 − □)) ÷ 36 = (4□ + 6) ÷ 36.
This equals 1/2 when 4□ + 6 = 18, that is, □ = 3. With 2 faces it is 14/36 and with 4 faces it is 22/36, so the game is not fair.
Because you get fixated on the numbers of red and blue faces and decide by intuition.
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