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Beat The Book (Posted on 2021-03-10) Difficulty: 3 of 5
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: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips Posting the Solution Soon | Comment 7 of 10 |
It's been a week since this puzzle was posted, I have the solution written up and will post it in another day or two, but wanted to see if anyone had any thoughts or wants to take another stab at a solution before I do.  

- I can confirm that my solution agrees with Charlie's computer calculations for part (a). Full points for a solution would involve proving it analytically.

- Part (b) can be accomplished by placing no more than 4 bets.



As an aside, this puzzle was based on a real conversation among coworkers, inspired by the relatively recent legalization of online sports betting in a number of US states. I added the simplifying assumptions in the puzzle to make a solution more concrete, and while they don't apply exactly as stated here in the real world, they're close enough - the basic strategy is valid and in practice it's straightforward to convert these "free bets" into real cash. 

  Posted by tomarken on 2021-03-17 09:35:28
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