Step by step
The question
Question: You roll a die.
You win $10 if you roll a 1 or 2.
Otherwise you lose $5.
What is the expected value?
1 Make a table
One row for each outcome: its value and its probability.
Win 10, with a chance of 1/3.
Lose: a loss is a negative value, minus 5.
The chance is 2/3.
The chances add up to 1: 1/3 and 2/3 make 3/3.
2 Multiply each row
Win: 10 times 1/3 is 10/3.
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 5 times 2/3 is minus 10/3.
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.
10 minus 10 is 0.
So the expected value is $0.
The expected value is 0: in the long run, nobody wins or loses.
That is called a fair game.
4 Check
Let's check.
Imagine 3 games.
The chances say: 1 win, 2 lose.
Together: we are up $0.
Shared over 3 games, that is $0 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.