Step by step
The question
Question: In a game, you win $20 with a 1 in 10 chance.
Otherwise you lose $3.
What is the expected value of the game?
1 Make a table
One row for each outcome: its value and its probability.
Win 20, with a chance of 1/10.
Lose: a loss is a negative value, minus 3.
The chance is 9/10.
The chances add up to 1: 1/10 and 9/10 make 10/10.
2 Multiply each row
Win: 20 times 1/10 is 20/10.
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 3 times 9/10 is minus 27/10.
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.
20 minus 27 is minus 7.
7 divided by 10 is 0.7.
So the expected value is minus 70 cents.
Negative: in the long run, you lose 70 cents a game.
4 Check
Let's check.
Imagine 10 games.
The chances say: 1 win, 9 lose.
Together: we are down $7.
Shared over 10 games, that is minus 70 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.