A nonprofit organization sells chances for a $40,000 automobile at $125 per ticket. It sells 2,000 tickets and offers four prizes, summarized in the table that follows. What are the expected winnings (or loss) for each ticket?

Question

A nonprofit organization sells chances for a $40,000 automobile at $125 per ticket. It sells 2,000 tickets and offers four prizes, summarized in the table that follows. What are the expected winnings (or loss) for each ticket? ​  ​

Quizplus · Accepted Answer

The expected value of one ticket can be calculated by multiplying the probability of each prize by its value and summing the products:

Expected value = (probability of winning prize 1 x value of prize 1) 
                   + (probability of winning prize 2 x value of prize 2) 
                   + (probability of winning prize 3 x value of prize 3) 
                   + (probability of winning prize 4 x value of prize 4)

The probability of winning one of the four prizes can be found by dividing the number of tickets sold by the number of possible outcomes (since each ticket corresponds to a unique 4-digit number between 0000 and 1999).

Probability of winning prize 1 (the car) = 1/2000
Probability of winning prize 2 (cash) = 3/2000
Probability of winning prize 3 (cash) = 4/2000
Probability of winning prize 4 (cash) = 192/2000

The values of the prizes are given in the table: 
Prize 1: $40,000 automobile
Prize 2: $2,500 cash
Prize 3: $1,000 cash
Prize 4: $125 cash (the cost of the ticket)

Plugging in the probabilities and values, we get: 
Expected value = (1/2000 x $40,000) + (3/2000 x $2,500) + (4/2000 x $1,000) + (192/2000 x (-$125))
Expected value = $20 - $4.6875 = $15.3125

Therefore, the expected winnings for one ticket are $15.3125, which is closest to choice letter B.

A Nonprofit Organization Sells Chances for a $40,000 Automobile at $125