What is a no-sweat bet worth
What odds should you take on a first-bet-refunded offer?
"Bet up to 1,000 dollars, refunded in bonus bets if it loses." The offer's value is not in the bet. It is in the refund, and you only collect the refund when the bet loses.
That inverts the usual instinct. The worth of the offer on its own is the refund, times how often you collect it, times what a bonus bet is really worth — here 75% after hedging:
| Qualifying price | Offer premium | Total EV |
|---|---|---|
| -110 | 375.00 | 329.55 |
| +150 | 450.00 | 450.00 |
| +300 | 562.50 | 562.50 |
| +600 | 642.86 | 642.86 |
The premium runs from 375.00 at a coin flip to 642.86 at a longshot — 1.71x more for taking the longer price. Bet the favourite and you win the bet most of the time, which is precisely how you fail to collect the thing you signed up for.
Two things this does not mean
It is not hedgeable. There is no second bet that locks a safety net, because the refund only exists in the branch where you lose. Any tool reporting this EV as guaranteed is reporting the wrong field.
Longer is not unboundedly better. Real longshots carry the heaviest margin, and past some point the extra vig costs more than the extra refund frequency earns. The right price is the longest one whose fair probability you actually trust.