Any Math Geniuses Out There?

[tripas for lunch]

Let's say a book is offering payouts on either side of an event:

For outcome a, the payout is A (1 unit stake plus winnings)

For outcome b, the payout is B.

I know that the house edge is as follows:

1/A +1/B - 1

This makes intuitive sense, but what I am looking for someone who can demonstrate a mathematical proof of why this is so.

[rfr3sh]

For A Spread Bet I would say:

Assuming that the line is 100% accurate then there should be a 50/50 chance that the team covers the number

so the fair odds should be +100/+100 (50/50)
however at -110 it is saying that there is a 52.38% chance that Team A or B covers the number
therefore the house has a 2.38 % edge

[Kaabee]

house edge on -110 is 4.55%. half the time we lose 110. half the time we win 100. so for every 220 bet we lose 10. 10/220 = .0454545