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## Multiple-Choice Lotteries

You have decided to spend \$100 on the lottery.

Is it better to buy 100 tickets for one week's drawing or one ticket each week for 100 consecutive weeks?

Does it matter as to the amount of the lottery winnings being advertised?

Source: Variation of a problem from Factorial, Summer 2001, p. 15

Hint: As Polya would say, try a simpler case. You have decided to spend \$10 on the lottery (assume you pick one of 100 numbers for each \$1). Is it better to buy 10 tickets for one week's drawing or one ticket each week for 10 consecutive weeks...knowing that the winning amount is \$100?

Or even, you have decided to spend \$2 on the lottery (again assume you pick one of 100 numbers for each \$1 and the winnings are \$100). Is it better to buy 2 tickets for one week's drawing or one ticket each week for 2 consecutive weeks?

Solution Commentary: The key is expected value. So, to detract you further, are you influenced by the fact that by the second option, you possibly could win every week, rather than just once?

To replicate the necessary analysis on a smaller scale, consider the dilemma of spending \$2 on the lottery (assume you pick one of 100 numbers for each \$1 and the winnings are \$100). You can either buy 2 tickets for one week's drawing or one ticket each week for 2 consecutive weeks?

For the first option, your expected winnings are (2/100)(\$100-\$2) = \$1.96. And, for the second option, the expected winnings are (1/100)(\$100-\$1)+(99/100)(1/100)(\$100-\$2) = \$1.9602.

Does this one simpler case convince you? Is the mathematical reasoning correct? Can you extend it to the original problem?