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