Step by step
The question
Question: In a game, you win $5 $5 with a 1 in 2 chance, win $1 with a 1 in 4 chance.
Otherwise you lose $10.
What is the expected value of the game?
1 Make a table
One row for each outcome: its value and its probability.
Win $5 5, with a chance of 1/2.
Win 1, with a chance of 1/4.
Lose: a loss is a negative value, minus 10.
The chance is 1/4.
2 Multiply each row
Win $5: 5 times 1/2 is 5/2.
We make the bottom 4: times 2, top and bottom.
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.
Win: 1 times 1/4 is 1/4.
Lose: minus 10 times 1/4 is minus 10/4.
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 plus 1 minus 10 is 1.
1 divided by 4 is 0.25.
So the expected value is 25 cents.
Positive: in the long run, you win 25 cents a game.
4 Check
Let's check.
Imagine 4 games.
The chances say: 2 win $5, 1 win, 1 lose.
Together: we are up $1.
Shared over 4 games, that is 25 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.