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counting and probability senior
Problem
A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins \1.\ If the number chosen does not appear on the bottom of either of the dice, the player loses \1.$ What is the expected return to the player, in dollars, for one roll of the dice? Give your answer as a fraction.
Solution
The probability of the number appearing 0, 1, and 2 times is respectively. So the expected return, in dollars, to the player is
Final answer
-\frac{1}{16}