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A game is played in which three dice are rolled, and the number of “1”s that appear is recorded. a) Determine the probability distribution for the random variable X, the number of “1”s in three rolls. b) Suppose you win $1 if a “1” appears once, you win $2 if a “1” appears twice, and you win $3 if a “1” appears three times. However, if there are no “1”s rolled, then you lose $1. Calculate your expected winnings/losses from playing this game. c) Is this game “fair”? Explain.
The probability of rolling a 1 on a die is 1/6 and the probability of not rolling a 1 on a die is 5/6. \[P(0\ 1s)=\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\] \[P(1\ 1)=\frac{1}{6}\times \frac{5}{6}\times \frac{5}{6}\] \[P(2\ 1s)=\frac{1}{6}\times \frac{1}{6}\times \frac{5}{6}\] \[P(3\ 3s)=\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}\]