The problem I want to solve is, given a free sports bet (i.e. if we lose we get our money back) and given no special knowledge of sports betting on the part of the player, what selection would have the most +EV odds to choose to bet on.

TL;DR part, skip to last paragraph if you want

Instinctively, if we bet our (e.g.) 10 dollar free bet at too low odds, say evens, even with a genuine 50-50 shot, we don't take the free money often enough and our EV is $5....
EV of loss = $0, probability of loss = about 0.5
EV of win = +$10 probability of loss = about 0.5
total = about +$5

With odds of 2/1 (+200 in US odds) and a 30% chance of winning, our EV is $6
EV of loss = $0, probability of loss = 0.7
EV of win = +$20, probability of win = 0.3 EV =$6

even though the bookie has vig in the second example, but not the first, our profit is higher because the free bet adds to our EV more often.

Over a certain point this factor will be less important (we're getting the free money almost all the time anyway) and the fact that "long shots never win", or rather they don't win often enough to justify their odds, will become more important.

J Dowie in "Efficiency and equity of betting markets" (1976), using data from the 1973 UK flat horseracing season gives the following figures showing how bookies' vig increases as odds increase (i.e. long shots are generally poor value).

Odds ...... winnings/wager % (i.e. percent of money returned to bettors as winnings)
Odds on ..... 99.2%
Up to 5/1 ... 90.0%
Up to 10/1 .. 89.4%
Up to 16/1 .. 80.3%
All ... 60.5%

Given that the EV of normal bets drops off at odds above 10/1, we are better off using our free bet at e.g. an 8/1 shot with only a 10% chance (EV $8), than an 11/1 shot with a 6.67% chance (EV $7.33).

So that suggests we are best off using our free bet on a selection with odds in the high single figures.

Hyung Song Shin in his paper
http://www.math.ku.dk/~rolf/teaching/thesis/Shin91.pdf

quotes Dowie and gives formulae to calculate real probabilities given the odds in a sporting event. It would be better to use this than the 1973 data, but I'm forced to admit that the maths is over my head until I see a worked example (my maths is like that, once I've seen it done with real numbers I can reproduce it, but not before).

end of TL;DR

Does anyone have access to a worked example of calculating Shin probabilities from the odds for a real sporting event?

If not then for example this weekend there is an English soccer match with odds of: Middlesbrough 1.4/1, West Ham 2.1/1 and Draw-Tie 2.2/1 Those probabilities add up to 105.17%, can someone show me how Shin would take the overround off to find the probabilities the bookies are working with in this simple 3-runner event?

in that paper it says "our model describes the market for bets surrounding a two-horse race". so that's only for 2 outcomes. also, the payouts are such that they pay 1$ on each horse (doesn't say what the stake is?).

anyway, if you have a real coinflip that offers 1.95 - 1.95 (50% chance for each), that's margin of 1.0256. you would get true odds by multiplying odds with this margin, so 1.0256*1.95 = 2. i'm not sure if you knew about this, if you did then this post won't help you much.

with your game, the logic is the same. odds are 2.4 - 3.1 - 3.2, margin 1.0517. so true odds would be 2.52 - 3.26 - 3.37 (with assumption that margin is the same on every outcome).

regarding your max EV bet for a freebet, i'm sure you should never take small odds (less than 1.8 or so) because there's rarely value in those. value is more likely to be found with higher odds (favourite bias). there's also a problem with high odds - their EV could be good, but since it's still unlikely to win, it could take a while for you to win a bet.

EV of your bet is: EV = chance of winning * (10*decimal odds - 10). so you can't really solve this problem in general, you just need to find a bet where you think the odds are higher than they should be. for the game in question, it would be best to bet on away team (EV = 1/3.37*(10*3.2-10) = 6.53$), second best choice would be draw (EV = 1/3.26*(10*3.1-10) = 6.44$) and the last for home team (EV = 1/2.52*(10*2.4-10) = 5.56$). but it's very important that the margin is the same on every outcome, which can be true for a game like this, but isn't for a game with odds 1.25 - 5.5 - 10.

forgot to write this: good idea with freebets is to bet both outcomes, for example back the draw with your freebet and then lay the draw on betfair or some other betting exchange. that way you can get around 70% of the freebet stake out with 100% certainty. if you want higher return than 70%, you would need to back and lay some other event at bigger odds, which would create a bigger liability.

Well this is the thing, if the margin is the same then I should just go for the longest odds (in the absence of any knowledge of the underlying sport) - but the idea of the paper (also confirmed by the 1973 horse racing results) is that there is a much higher margin on the long shots as bookies need a higher risk premium to lay the bets when sharps could potentially take them to the cleaners.

for the footy game you mentioned, both teams are pretty even, 3.2 isn't exactly a longshot, so i think margins should be about the same overall, for high odds (say >5) this doesn't hold anymore.

here's a calculator for what i was talking about (hedging a freebet). http://bookiescash.com/tools/risk-fr...alculator.html

almost all freebets are of the "stake not returned" type. back commission 0%, lay commission maybe 5 or 7%, depending where you live and how exactly it's calculated (some exchanges take it both on winning and losing bets, some only on winning).

for example, with your game if you backed one team at 3.2 and layed on betfair at 3.35 with 5% commission, you'd have a return of 63.3% from the freebet stake. if i had a bunch of freebets, i would hedge them like this.

I have a 5 euro free bet per day till the 29th, but I don't need to hedge them, I'm happy enough to let them ride when it's a relatively small amount.

To get the free bet I have to make 4 normal bets also for 5 euros per day, and a related problem is also how to minimize the EV loss over those 4 bets - the above wall of text suggests the shorter odds will be best, and it's actually a condition of the offer that the odds be at least 1.4 in decimal odds (2/5 on or -250 in US odds), so they obviously think that too. On the other hand, in soccer, the markets are known to tend more towards giving better value for the draw/tie as it's not a popular bet with recreational punters, so I've been betting a lot of football draws. Also just whatever's on at the time I log in.

In any case, I'm pretty sure the offer is plus EV for me.

yeah, it will be EV+. you can easily check that just through the hedging calculator. your loss with hedging a qualifying bet should be on average 0.4$. so, -0.4*4 = -1.6$. that's the cost of getting a freebet. then with a freebet you profit around 3$, which gives 1.4$ profit overall per one day of such bets.

but you'll have variance with straight betting, which isn't really a problem with these amounts.