An online sportsbook is advertising a “Risk Free” promotion for new users: If your first wager loses, you will be awarded a “free bet” in the amount of your wager (up to $1,000).

In reading the terms and conditions of the promotion, you note that unlike a normal wager, a “free bet” does not return the stake along with any winnings. e.g. if you wager $1,000 cash on a selection with odds of 1.95 and win, you’d get back $1,950 (your $1,000 stake is returned, plus $950 in winnings); if you’d placed a $1,000 “free bet” on that same selection, you’d only be paid the $950 in winnings.

For this puzzle make the following simplifying assumptions:
- The sportsbook reduces the fair odds of all of their selections by 2.5%. For example, the true odds on selection with a 50% probability of occurring should be 2.00. However the sportsbook pays only 2 * (1 - 0.025) = 1.95.
- The sportsbook knows the true probabilities of all events occurring, and uniformly prices all selections with the same 2.5% margin.
- The sportsbook offers a large enough variety of markets that any odds you seek are available for you to bet on.

a) Determine a strategy to maximize the expected value of this promotion.

b) Assuming you wanted to make this a truly risk-free proposition, maximize the amount of guaranteed profit you can get out of this promotion.

(In reply to Guaranteed Profit (spoiler) by Steve Herman)

You have the right general idea but you only receive a free bet if your first wager (based on time of bet placement) loses.  Any subsequent losing bets do not receive a free bet.  

Posted by tomarken on 2021-03-17 12:01:17