implied probability vs house edge

Say a sportsbook is offering a market on a fair coin flip at -110 for heads and -110 for tails.

We know how to calculate the house edge for betting heads:

win -- receive 110+100 = 210 * 0.5 = 105
lose -- receive 0 * 0.5 = 0

110 bet - 105 expected return = -5.

-5/110 = -4.545 percent.

This is confirmed here -- https://wizardofodds.com/games/sports-betting/straight-bet-calculator/ -- by entering -110 in each box.

Now we calculate the implied probability of each option:

stake/payout = implied probability

heads: 110/210 = 52.381%
tails: 110/210 = 52.381%

How do we convert implied probability to house edge?

If we add up 52.381 and 52.381 we get 104.762 - 100 = 4.762%

How do we go from this number -- 4.762% -- to the correct house edge of 4.545%?

What small piece of algebra am I missing here?

104.762 is not 100.

100/104.762 X 4.762 = 4.545.

Thanks eljefe,

Your formula works perfectly.

I just tried it with:

-800
+400

= 108.89 implied probability

then

(100/108.89) * (108.89 -100) = 8.164 percent, which is the correct balanced house edge.

Would you mind explaining the reasoning or how you got this conversion?

And does anyone know WHY the balanced house edge is always slightly less than the sum of the implied probabilities - 100? Many websites (apparently incorrectly) say the house edge in a moneyline market is simply the sum of the implied probabilities -100.

OK, I think I've got it.

The house edge is (the sum of implied probabilities - 100) / (the sum of implied probabilities)

When using bookmakers with odds on various outcomes (horses, teams winning matches etc.) there's a concept of "book value" and is based on how much you need to wager on each outcome to guarantee $100 back. This will usually give a number over $100 and in your example you add $52.381 (10/11 or -110) twice to get $104.762. This means the bookmaker makes $4.762 for every $104.762 you bet which is the same as making $4.545 on every $100. So the book value is 104.762 and the House Edge 4.545.

An easier example is if there were three horses: Evens, 2/1 and 3/1. It's fairly simple to see that to guarantee $12 you have to place 6, 4 and 3 = $13; so the book value = 108.333. Similarly the bookie makes $1 for every $13 wagered = 1/13 = 7.692% House Edge.

btw I did see a race at Windsor (UK) that had that exact Starting Price.

The difference is the book value is how much extra you have to bet to receive $100, the House Edge is how much less you get back (i.e. lose) if you bet $100.

Quote: sodawater

Many websites (apparently incorrectly) say the house edge in a moneyline market is simply the sum of the implied probabilities -100.

That’s close but not exact. If the sum of the implied probabilities is 1.08, then the edge is .08/1.08 = 7.4% not 8 %

1.08 means you’d have to bet much that to guarantee a return of 1.00