What is the expected value of this question?

Question

The daily number lottery costs $1 to play. Bella must pick two digits in order from 0 to 9 and duplicates are allowed. If she wins, the prize is $80. First calculate the expected value of the lottery. Determine whether the lottery is a fair game. If the game is not fair, determine a price for playing the game that would make it fair.

Answer 1

The chances of winning are 1/100 because there are 100 combinations of 2 digits 0-9 including duplicate digits. Since the cost of play=$1, there would be a 0.01 chance of winning $80, which is a net gain of 80-1=$79.

The chance of losing is 0.99 and the net loss is $1 (the cost of play), represented by -$1.

Combine these: 0.01×79-0.99×1=$(0.79-0.99)=-$0.20, which is the value of the lottery entry.

So, because the value is negative, the lottery is not fair.

The lottery would be fair if the value of play=0. If $p is the cost to play then:

0.01×(80-p)-0.99×p=0.8-0.01p-0.99p=0.8-p=0 so p=$0.80 or 80c.

CHECK: 0.01×79.20-0.99×0.80=0.7920-0.7920=0.

For a fair lottery the cost of play should be 80 cents.