Question

Miguel is playing a game in which a box contains four chips with numbers written on them. Two of the chips have the number 1, one chip has the number 3, and the other chip has the number 5. Miguel must choose two chips, and if both chips have the same number, he wins $2. If the two chips he chooses have different numbers, he loses $1 (–$1).

What value should be assigned to choosing two chips with the number 1 to make the game fair? Explain your answer using a complete sentence and/or an equation.
What value should be assigned to choosing two chips with the number 1 to make the game fair? Explain your answer using a complete sentence and/or an equation.

Answer 1

Explanation:

4 chips in total.

2 chips have a 1.

1 chip has a 3.

1 chip has a 5.

to get 2 chips with the same number it must be the chips with the number 1.

the text does not say, but I assume that after pulling the first chip it is not put back into the box for the second pull.

the probabilty to have a 1 in the first pull is

2/4 = 1/2

then the probabilty to pull the remaining 1 in the second pull is

1/3

as there are only 3 coins left for the second pull.

so, the probabilty to pull 2 times a 1 is

1/2 × 1/3 = 1/6

that automatically means that getting 2 different numbers (which is anything else than getting the two 1s) has the probability

1 - 1/6 = 5/6

"to be fair" as I understand it means that the expected value at the end of e.g. 6 attempts is a simple $0.

it would mean we expect in 6 attempts to get 5 times a result with 2 different numbers with -$1 each. that is -$5 in total.

and we expect one time to get 2 equal numbers.

in order to make it a fair 0, this case has to compensate for the other -$5. and so, the winning for having two equal numbers must be $5.

and then the sum of -$5 and $5 is the fair $0 balance.

so, the winning value for getting 2 equal numbers should be $5 to be totally fair.

Answer 2

The value that should be assigned to choosing two chips with the number 1 to make the game fair is $0.50. The expected value of winning is (2/6) x (1/3) x $2 = $0.11. The expected value of losing is (4/6) x (2/3) x (-$1) = -$0.44. The expected value of playing the game is $0.11 - $0.44 = -$0.33. To make the game fair, the value of winning must be equal to the value of losing, so the value of choosing two chips with the number 1 should be $0.50.