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.
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