# how to calculate expected value?

Imagine a lottery with 400 tickets. There are equal chances(100 tickets each) of winning 0 dollars, ten dollars, 50 dollars and 100 dollars. The tickets are being sold for 60 dollars. what is the expected value of buying a ticket?

Relevance

For each prize value, find the probability of each prize occurring.

Then multiply each prize, minus the cost of the ticket, by that probability, and add the products together.

So if there are 100 tickets each of \$0, \$10, \$50, and \$100 winners, then the probability of each prize is 1/4.

Subtracting out the \$60 cost for each ticket, find the net result after each prize is awarded to get:

0 - 60 = -60

10 - 60 = -50

50 - 60 = -10

100 - 60 = 40

Finally, multiply those amounts by their probabilities and add them together:

(1/4)(-60) + (1/4)(-50) + (1/4)(-10) + (1/4)(40)

-60/4 - 50/4 - 10/4 + 40/4

The denominators are the same so add the numerators:

(-60 - 50 - 10 + 40)/4

and simplify:

-80/4

-20

The expected value per play is a \$20 loss.