Step by step
The question
Question: You roll a die.
You win $6 if you roll a 6.
Otherwise you lose $1.
What is the expected value?
1 Make a table
One row for each outcome: its value and its probability.
Win 6, with a chance of 1/6.
Lose: a loss is a negative value, minus 1.
The chance is 5/6.
The chances add up to 1: 1/6 and 5/6 make 6/6.
2 Multiply each row
Win: 6 times 1/6 is 6/6.
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 1 times 5/6 is minus 5/6.
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.
6 minus 5 is 1.
1 divided by 6 is 0.167, and more digits.
To the nearest cent, the expected value is 17 cents.
Positive: in the long run, you win about 17 cents a game.
4 Check
Let's check.
Imagine 6 games.
The chances say: 1 win, 5 lose.
Together: we are up $1.
Shared over 6 games, that is about 17 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.