Discrete probability

Question

The student council is hosting a drawing to raise money for scholarships. They are selling tickets for $9 each and will sell 900 tickets. There is one $2,000 grand prize, three $200 second prizes, and twelve $10 third prizes. You just bought a ticket. Find the expected value for your profit. Round to the nearest cent.

Answer 1

Hello Chantal,

In this problem the value of the random variable x is your profit and P(x) is the probability of each profit outcome. So that if you just bought a ticket for $9 your possible profits and corresponding probabilities are as follows,

Grand prize: 2000-9=1991 P(1991)=1/900 (since there is only one grand prize winning ticket)

Second prize: 200-9=191 P(191)=3/900 (since there are only three second prize winning tickets)

Third prize: 10-9=1 P(1)=12/900 (since there are only twelve third prize winning tickets)

Lose: -9 P(-9)=884/900 (since there are 900-1-3-12=884 losing tickets)

The expected value of a discrete random variable, also called the mean is the sum of the products of each value and it's corresponding probability.

Hence,

The expected profit is 1991*(1/900)+191*(3/900)+1*(12/900)+(-9)*(884/900) = -5.98

You can expect to lose $5.98 per ticket on average.