Question

A simple game goes as follows: you pay one dollar and roll a die. If the roll is either a 1 or 2, you get your dollar back. If it is greater than 3 you lose the dollar you paid. If it is a 3, you get your dollar back and an additional dollar. What is the expected value of the net amount of money you win/lose?

Answer 1

-1/3

Explanation:

Given that a simple game goes as follows: you pay one dollar and roll a die. If the roll is either a 1 or 2, you get your dollar back. If it is greater than 3 you lose the dollar you paid. If it is a 3, you get your dollar back and an additional dollar.

Your fee for the dollar would be 1 dollar always

If you roll 1 or 2, your gain is 0

If you roll 3, your gain is 1

If you roll 4 or 5 or 6you lose 1 dollar

Let X be the amount of gain.

X 0 1 -1

P 2/6 1/6 3/6

E(x) = Sum of products of x with probabilities

=
`(1)/(6) -(3)/(6) \\=(-2)/(6) \\=(-1)/(3)`

Note: this is not a fair game to play since expected gain is negative.