An unsavory individual offers a game for you to play. You roll a single die, and if the roll is 3 or higher, you win $2. If you roll a 1 or a 2, you owe him a dollar. Little do you know, the die is loaded. The probability that you'll roll a 1 is 0.5, and the rest of the values have equal probabilities. What is the expected value of this game

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LammettHash LammettHash · Answer 1 · 2019-08-22T19:09:35+02:00

The probability of rolling a 1 is 0.5, so the remaining five possible outcomes (2-6) each have probability 0.1.

If [tex]W[/tex] represents your winnings from the playing the game, then

[tex]P(W=w)=\begin{cases}0.6&\text{for }w=2\\0.4&\text{for }w=-1\end{cases}[/tex]

because the probability of rolling a 1 or 2 is

[tex]P(1\text{ or }2)=P(1)+P(2)=0.5+0.1=0.6[/tex]

and rolling any of the other values is complementary to this event.

The expected value of the game is then

[tex]E[W]=\displaystyle\sum_wwP(W=w)=2\cdot0.6+(-1)\cdot0.4=\boxed{\$0.80}[/tex]