Question

Four possibly winning numbers for a lottery—AB-4536, NH-7812, SQ-7855, and ZY-3221—
arrive in the mail. You will win a prize if one of your numbers matches one of the winning
numbers contained on a list held by those conducting the lottery. One first prize of $100,000,
two second prizes of $50,000 each, and ten third prizes of $1000 each will be awarded. To be
eligible to win, you need to mail the coupon back to the company at a cost of 33¢ for postage.
No purchase is required. From the structure of the numbers that you received, it is obvious the
numbers sent out consist of two letters followed by four digits. Assuming that the numbers
you received were generated at random, what are your expected winnings from the lottery? Is
it worth 33¢ to enter this lottery?

Answer 1

We first of all calculate our sample size. The number of ways of generating two letter and four digits in a way that repeatition is allowed is:
26^2 x 10^4 = 676 x 10000 = 6,760,000.
Therefore, there are 6,760,000 possible number combinations for the lottery.

Probability of winning the first prize = 1/6,760,000
Probability of winning the second prize is 2/6,760,000 = 1/3,380,000
Probability of winning the third prize is 10/6,760,000 = 1/676,000
Probability of winning nothing is (6,760,000 - 13)/6,760,000 = 6,759,987/6,760,000                 *[there are 13 winnings in all]

Expected winning from the lottery = (1/6,760,000) x (100,000 - 0.33) + (1/3,380,000) x (50,000 - 0.33) + (1/676,000) x (1,000 - 0.33) + (6,759,987/6,760,000) x (-0.33) = 0.0148 + 0.0148 + 0.0015 - 0.3300 = -0.30

Therefore, the expected winning is a loss of 30 cents.
This means that it is not worth 33 cents to enter this lottery.