Step by step
The question
Question: It costs $2 to play.
You roll a die and win $6 if you roll an even number.
Is the game fair?
Find the expected value.
1 Make a table
One row for each outcome: its value and its probability.
Win: the prize is $6, but we paid $2.
So we are up $4.
The chance is 1/2.
Lose: a loss is a negative value, minus 2.
The chance is 1/2.
The chances add up to 1: 1/2 and 1/2 make 2/2.
2 Multiply each row
Win: 4 times 1/2 is 4/2.
The expected value: each value times its probability, all added up.
It is what we win or lose per game, on average, if we play many times.
Lose: minus 2 times 1/2 is minus 2/2.
3 Add them
Same bottoms: we add the tops.
A loss takes away.
4 minus 2 is 2.
Both 2 and 2 can be divided by 2.
That makes it simpler.
So the expected value is $1.
Positive: in the long run, you win $1 a game.
A fair game has 0.
So this game is not fair.
4 Check
Let's check.
Imagine 2 games.
The chances say: 1 win, 1 lose.
Together: we are up $2.
Shared over 2 games, that is $1 a game.
The same.
So the answer is right!
Let's look back.
First: make a table.
Then: multiply each row.
Then: add them.
Last: check.