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Agree or Disagree?

On behalf of her family and friends, Frances Bobner threatened to sue the Pennsylvania Lottery Commission in 1994.

Background: Over a ten year period, they had spent \$150,000 annually on lottery tickets...with no winning tickets.

Her claim: If the lottery system was fair in line with its stated odds, their tickets should have produced some winners....and some financial return.

Problem: What are your thoughts on this? Should she have won something? Obviously, her expected winnings will depend on the game played.

Suppose it was their Big Four Lottery game, where, on each play, you pick a four-digit number (e.g. 2374)...and your bet type: straight or box. On a straight wager, you must match the winning digits in the exact order (e.g. 2-3-7-4), while in a box wager, you can match the winning numbers with the digits in any order (e.g. 3-4-2-7).

For the straight wager, the odds are 1 in 10,000, with payout of 5000 to 1. For the box wager, the odds are 24 in 10,000, with payout of 200 to 1. Also, on each ticket, she could bet \$0.50, \$1, \$2, \$3, \$4, or \$5. Thus, with a bet of \$0.50, a winning ticket would return \$2500 (staight wager) or \$100 (box wager).

Suppose she spent \$411 daily to play this game using \$0.50 bets on the Box option. That is, 822 games a day for 10 years.

Problem: Given this scenario, what is her expected return? Should she expect to have had at least one winning ticket over the 10 years?

Final Note: I have done a quick search of the Pennsyvania court records, and have not found any record that Frances Bobner ever filed the claim....Almost 20 years have passed since her threat, but...!

Source: Adapted from J. Cohen's "She Hopes for a Lotto Legal Luck," National Law Journal, May 9, 1994.

Hint: Review what expected value means and how it is calculated.

For example, suppose you bet \$1 on the roll of a fair 6-sided die, with a win (even prime number rolled) producing \$4. What is your expected winnings? Over 1000 games involving total bets of \$1000, what should be your expected gain or loss?

Solution Commentary: I will let you argue it out...