Step by step
The question
Question: A raffle sells 50 tickets at $4 each.
There is one prize of $100.
What is the expected value of one ticket?
1 Make a table
One row for each outcome: its value and its probability.
Win: the prize is $100, but we paid $4.
So we are up $96.
The chance is 1/50.
Lose: a loss is a negative value, minus 4.
The chance is 49/50.
The chances add up to 1: 1/50 and 49/50 make 50/50.
2 Multiply each row
Win: 96 times 1/50 is 96/50.
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 4 times 49/50 is minus 196/50.
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.
96 minus 196 is minus 100.
Both 100 and 50 can be divided by 50.
That makes it simpler.
So the expected value is minus $2.
Negative: in the long run, you lose $2 a ticket.
4 Check
Let's check.
Imagine 50 tickets.
The chances say: 1 win, 49 lose.
Together: we are down $100.
Shared over 50 tickets, that is minus $2 a ticket.
The same.
So the answer is right!
Let's look back.
First: make a table.
Then: multiply each row.
Then: add them.
Last: check.