Step by step
The question
Question: It costs $2 to play.
You roll a die and win $9 if you roll a 6.
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 $9, but we paid $2.
So we are up $7.
The chance is 1/6.
Lose: a loss is a negative value, minus 2.
The chance is 5/6.
The chances add up to 1: 1/6 and 5/6 make 6/6.
2 Multiply each row
Win: 7 times 1/6 is 7/6.
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 5/6 is minus 10/6.
3 Add them
Same bottoms: we add the tops.
A loss takes away.
Careful!
The chances are not equal.
Each value counts as much as its probability.
7 minus 10 is minus 3.
Both 3 and 6 can be divided by 3.
That makes it simpler.
So it is minus 1/2.
1 divided by 2 is 0.5.
So the expected value is minus 50 cents.
Negative: in the long run, you lose 50 cents a game.
A fair game has 0.
So this game is not fair.
4 Check
Let's check.
Imagine 6 games.
The chances say: 1 win, 5 lose.
Together: we are down $3.
Shared over 6 games, that is minus 50 cents 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.