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 Beat The Book (Posted on 2021-03-10)
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.

 See The Solution Submitted by tomarken Rating: 4.5000 (2 votes)

Comments: ( You must be logged in to post comments.)
 Subject Author Date re: Guaranteed Profit (spoiler) tomarken 2021-03-17 12:01:17 Guaranteed Profit (spoiler) Steve Herman 2021-03-17 10:35:15 Posting the Solution Soon tomarken 2021-03-17 09:35:28 re: Possible Solution tomarken 2021-03-12 08:29:06 Possible Solution broll 2021-03-10 20:17:24 re: Almost Guaranteed profit? tomarken 2021-03-10 18:44:23 re: Almost Guaranteed profit? Charlie 2021-03-10 18:33:30 Almost Guaranteed profit? Steve Herman 2021-03-10 18:21:05 computer calculations Charlie 2021-03-10 12:39:51

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