Step by step
The question
Question: In a game, you win $8 with a 1 in 4 chance.
Otherwise you lose $2.
What is the expected value of the game?
1 Make a table
One row for each outcome: its value and its probability.
Win 8, with a chance of 1/4.
Lose: a loss is a negative value, minus 2.
The chance is 3/4.
The chances add up to 1: 1/4 and 3/4 make 4/4.
2 Multiply each row
Win: 8 times 1/4 is 8/4.
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 3/4 is minus 6/4.
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.
8 minus 6 is 2.
Both 2 and 4 can be divided by 2.
That makes it simpler.
So it is 1/2.
1 divided by 2 is 0.5.
So the expected value is 50 cents.
Positive: in the long run, you win 50 cents a game.
4 Check
Let's check.
Imagine 4 games.
The chances say: 1 win, 3 lose.
Together: we are up $2.
Shared over 4 games, that is 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.