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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.)
Solution Guaranteed Profit (spoiler) | Comment 8 of 10 |
(In reply to Posting the Solution Soon by tomarken)

Oh, I think I got it, with that hint.  


Place 4 $1000 bets on the flip of a single coin, 2 bets on heads and 2 bets on tails.  You will win 2 bets and lose 2 bets, leaving you with $3900 and 2 "free bets".

Use the "free bets" to bet on another flip of the coin, 1 bets on heads and 1 bet on tails.  You will win one of the bets, for another $975.

Total guaranteed receipt = $4875 against a $4000 outlay. 

  Posted by Steve Herman on 2021-03-17 10:35:15
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