Algebra / Probability Genius Required

[HiLo]

Can one of you maths geniuses help me solve this problem:

I am being offered decimal odds on 4 possible outcomes of a race @
1.79, 5.48, 9.35 & 6.35

I can work out the implied probabilities from these odds as follows:
runner A: 1.79 = 55.86%
runner B: 5.48 = 18.25%
runner C: 9.35 = 10.69%
runner D: 6.35 = 15.75%

The total implied probability comes to 100.55% showing a slight advantage to the bookmaker and thus preventing me from using a dutching system to make this +EV.

However, I earlier placed a $2 bet on runner A at odds of 2.6 (showing a $3.20 profit if this runner wins). This means I can now use a dutching system to make this +EV in all 4 outcomes. (I threw some bets into the spreadsheet until I came up with A=18, B=7, C=4 & D=6 which appears to work.)

What formula should I use to work out the best bet sizes to extract most profit from this situation???

(I am guessing it is something to do with dividing the $3.20 by the .55 overround but am getting lost somewhere....)

[spoonitnow]

What the fuck is dutching? This is interesting to me.

[HiLo]

dutching is where you place a bet on every runner with the stakes adjusted so that all outcomes show a profit. (Named after Al Capone's accountant reputedly)

In order to successfully apply a dutching system the total implied probability must be <100

[spoonitnow]

dutching is where you place a bet on every runner with the stakes adjusted so that all outcomes show a profit. (Named after Al Capone's accountant reputedly)

In order to successfully apply a dutching system the total implied probability must be <100

Gotcha.

[Galapogos]

Dutching sounds dirty.

[mrickett]

wow I just finished my probability and statistics course 3 months ago and already this makes no sense. The second I left that class room for the last time I left my knowledge there!

[biondino]

Is the technical term "arbitrage"?

[Numbr2intheWorld]

sorry i don't know.

[euphoricism]

Is the technical term "arbitrage"?

Basically the same thing, I think. Betting both sides with a gaurenteed +EV result as in sportsbetting.

[HiLo]

Thanks to those who showed an interest:
I have since confirmed that you can't increase the equity by backing runner A at the new odds.

If I wish to guarantee a win by dutching the other 3 selections that is fine, but the return will always be limited by the earlier bet.
i.e. bet sizes should be:
B = (5.2 / 5.48) = 0.95
C = (5.2 / 9.35) = 0.96
D = (5.2/6.35) = 0.82
producing a guaranteed return of $0.88 on any outcome.

If i add to runner A at the new odds then dutch it across the other 3 I will return less than $0.88.

Wish I could express this in terms of algebra though...

[swiggidy]

I never tried to do the math because I haven't been that bored.

Intuition tells me that if the sum of the odds is > 100% it doesn't really matter how you spread the money profit wise. You could minimize variance, but eV is typically eV in a non-decision based environment.