Step by step
The question
Question: In a game, you win $10 with a 1 in 4 chance.
Otherwise you lose $2.
What is the expected value of the game?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Make a table
One row for each outcome: its value and its probability.
Win 10, 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: 10 times 1/4 is 10/4.
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!
We cannot just take the middle of the values: they do not happen equally often.
Each value counts as much as its chance.
10 minus 6 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, 3 lose.
Together: we are up $4.
Shared over 4 games, that is $1 a game.
The same.
So the answer is right!
5 Pick the choice
Our answer is E = $1.
That is choice A.
6 Why not the others
B is $2.
Wrong: used 1/4 for losing too: losing has probability 3/4.
C is $2.50.
Wrong: forgot the losing part.
D is $8.
Wrong: 10 − 2: left out the probabilities.
Let's look back.
First: make a table.
Then: multiply each row.
Then: add them.
Then: check.
Then: pick the choice.
Last: why not the others.