House edge calculation

How do you calculate house edge?

For example the probability of the first card from a single deck being any ace is 12 to 1 or 7.692% (I think)

If you offered odds of 10 to 1 or 11 to 1 how would you calculate what your hose edge would be?

For something simple like that, you simply add the return on all possible outcomes * the probability of each occurring. So in your example, if it paid 11-1, you'd calculate it this way:

For a $1 bet...
(1/13)*(1*11) + (12/13)(-1)
11/13 - 12/13
-1/13
For every $1 bet, you're expected to lose $1/13, or $0.07692 (7.692 cents).
So divide the expected loss by the amount bet to get the house edge...
0.07692/1 = 7.692% house edge

Thanks for the reply. Is there another way using the percentage? ie If you offered 20 to 1 on an outcome which had a 3.582% chance?

Quote: ajb268

Thanks for the reply.
Is there another way using the percentage? ie If you offered 20 to 1 on an outcome which had a 3.582% chance?

https://wizardofvegas.com/forum/gambling/tables/1213-variance-in-craps/

For a wager that pays x:1 with a probability of winning p, the house edge and standard deviation (per unit bet and per 'root decision') are:

he = ((x+1)*p) - 1

std = (x+1)*Sqrt[p*(1-p)]

Thanks again and please forgive my ignorance but shouldn't the house edge decrease as the odds offered (x) increase?

Quote: ajb268

Thanks again and please forgive my ignorance but shouldn't the house edge decrease as the odds offered (x) increase?

I know what it is like to stare at mathematical expressions like they were hieroglyphs from Mars, but when you can just plug in values, just do so.

he = ((x+1)*p) - 1

test that against something known, to make sure we understand it.
we know any 7 has an HE of 16.67%, has 1 in 6 chance of being rolled, pays 4:1

use http://web2.0calc.com/ and you can just copy and paste into it. Will even preserve algebraic expression.

he = ((x+1)*p) - 1 = ((4+1)*1/6) - 1 = -0.16666666666667 = looks like that is working [g]

still looks like hieroglyphs but all I did was plug in numbers


>>>

the probability of the first card from a single deck being any ace is 12 to 1 [or 1/13]

If you offered odds of 10 to 1

he = ((10+1) * 1/13) - 1 = (11/13) -1 = -0.15384615384615
_

If you offered odds of 11 to 1

he = ((11+1)*1/13) - 1 = -0.07692307692308

the HE decreases from about 15.4% to about 7.7%

PS: looks like 24 bingo is right, if that was the issue. I seem to readily convert to 'absolute value' without hardly noticing it.

Quote: ajb268

Thanks again and please forgive my ignorance but shouldn't the house edge decrease as the odds offered (x) increase?

It should have been 1 - (x+1)*p.

(But welcome to his ignore list for daring to question him.)