Problem 180SE

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:

Step 1 of 1:

Four winning numbers for a lottery is which arrived in the mail. You will win a prize if the number matches one of the winning numbers on a list held by those conducting the lottery.

One first prize worth of , two second prizes worth of each, and ten third prizes of $1000 each will be awarded.

To be eligible to win, you need to mail the lottery back to the company at a cost of 33 cents for postage.

Assuming that the numbers received were generated at random.

What is the expected winnings from the lottery, is it worth 33 cents to enter this lottery?

Let be the random variable which represents the winning of the lottery.

Hence we need to find the

Possible combinations of letters and numbers for the lottery number is,

Since an alphabet in English contains 26 letters and the number of digits in numeric system is 10, hence choosing one letter out of 26 letters is 26 ways, likewise choosing one digit out of 10 digits is 10 ways

Thus, the probability of winning a prize if one of your numbers matches with one of the winning numbers which have 676000 possible combinations.

Hence,

……….(1)

……….(2)

……….(3)

The expected value of any random variable can be written like this,

Hence the expected winnings from the lottery is,

Hence the expected winnings from the lottery are

Since the expected winnings from the lottery are which is much less than the price of the stamp, hence it is not worth to enter this lottery.