Probability / Gambling question

Factual Questions
1

This is probably a fairly simple question for any mathematics people out there:

At the start of the rugby world cup I put a $40 bet on Australia to win, at odds of $5. (In other words, I will receive $200 if they win, for a profit of $160.)

Australia are in the final now vs England. England are slight favourites to win: here in New Zealand they are paying $1.70 to Australia’s $2.00.

I now want to put a bet on England: how much should this bet be, assuming I want to maximise my expected return? (I’m not sure what % these odds equate to, I would guess that England is given about a 60% chance of winning.)

PS Yes I am a trecherous NZer for betting on the Aussies; what can I say, I followed my head not my heart.

2

If you want to maximize your expected return, then the answer is simple: If you think the odds of England winning are better than the odds being offered, bet as much as you can. The prior bet is irrelevant to the expectation of making a new one.

3

Yes to maximize your winnings have $160 on England. At $1.70 an England win gets you $272, all profit. The most you can win if Australia win is $160 by not saving on England. Personally my bet on the final will be the $10 about no try scored.

4

I think the question you are asking is how much you should bet to maximise your winnings regardless of the result.

Therefore:

If Aus win you get $200 - $newbet
If England win you get $newbet * 1.7

Therefore if you bet £74 you win $126 if Aus win and $125.8 if England win. This doesn’t include your original $5 stake which I assume you’ve written off now anyway.

Someone please check my maths :wink:

5

Fez you’re getting closer to what I was asking, I’m just not sure you’re quite right. I want to take into account the fact that England are more likely to win this match, ie I want to assume that the posted odds are correct. That’s not something that you or the first two posters have considered and it’s something I think should be, if I’m going to do this properly.

6

I’ve reread your post and realised that your original stake was $40 (actually quite significant), therfore:

If Aus win you get $200 - $newbet
If England win you get ($newbet * 1.7) - $40

Therefore if you bet £89 you win $111 if Aus win and $111.3 if England win.

Someone please check my maths (again)

7

I posted that last one before you replied (must use preview!!)

Using the bookies odds of 1.7 then they think that there is a 59% chance of england winning and therefore 41% change of Aus winning (ignoring the chance of a draw and the fact that they have a profit margin in there).

Using my earlier calculations but multiplying the england winnings by 0.41 and the Aus winnings by 0.59, it skews the optimum stake to $104.
Therefore you win $96 if Aus win and $136.8 if England win.

8

From a bookmaker’s standpoint, the odds of either team winning is not what they are focusing on, but rather their estimate of bet popularity. So you can’t infer the actual winning odds on the contest itself.

The bookmaker wants to make a profit whoever wins and they do not wish to become a rooter for either side. Of course, when they post payouts they run the risk of not estimating correctly the proportion of money betted on each team.

The $1.70, $2.00 indicates they believe for each $1 betted on the more popular team, 71-99 cents will be betted on the less popular one. (They may also be taking into account the bets and odds established previously).

9

don’t ask and fezpp: note from the OP that the stated odds are 2:1.7 in favor of England, not 1:1.7 for England (fezpp) or 1:1.7 against England (don’t ask).

Trillionaire: To maximize your return, make no further bets, if you are assuming the bookie’s odds to be accurate (keep aahala’s comment in mind when deciding whether they are – I’ll assume they are for the purposes of this post).

At present, you have $40 of unfair money invested in this game. It is unfair because you have it paying at 1:5 odds, when the true odds are 1.7:2. That is quite unfair in your favor.

Any further money you add into the system will be fair money – the payout odds will match the true odds. For every dollar of fair money you add into the system, your total investment becomes more and more fair, meaning you lose more and more of your edge.

2:1.7 odds means that England has a 54% chance of victory. Your current expected outcome is then:

(54% * $0) + (46% * $200) - $40 = $52

That’s (odds of England victory * return for England victory) plus (odds of Aussie victory * return for Aussie victory) - (initial investment).

Now let’s add some cash on England, say, $10. The payout is 1.7:2, so if England wins, you get $8.50. Your total expected return is then:

(54% * $8.5) + (46% * 200) - $50 = $46.59.

Adding $10 to the Aussies doesn’t help either:

(54% * $0) + (46% * 211.76) - $50 = $47.40.

The way to maximize your return is to make unfair bets: bets with payout odds better than the true odds. So, if you had accurate information that England is actually 60% likely to win the game, instead of the 54% predicted by the bookies, you should bet, bet, bet on England. As Same Stone mentioned, this bet would be entirely unaffected by any previous bets. If you don’t have any such information, your original bet has already turned out to be a good bet; don’t water it down by making even-money bets.

10

Trillionaire

could you clear up a slight confusion for me?

you say you bet 40 at odss of 5, yet you are expecting $200 back for a net profit of $160.

At 5- 1 would you not get $240 dollars back? the $200 winnings & the original stake?

11

One and Only the odds are expressed differently here in New Zealand from many other places. My bet on Australia is at $5.00, which equates to odds of 4:1. Aussie are currently paying $2.00, which is even money (1:1). England are paying $1.70, which is the same as 7:10 (I think.)

And thanks for your replies everyone.

12

I don’t think you want to know about probability, you want to know about hedging. Math isn’t really my thing, and it looks like other people took care of that anyway. I just wanted to clear up the terminology.

13

Maybe do a google search on “dutching the books,” which is what you’re talking about here. Basically, dutching the books involves making a bet on Team A to beat Team B when the odds favor Team A, then making a bet on Team B when the odds switch to favor Team B, and if you bet the correct amount then you win money no matter who wins because you’re taking advantage of the different odds given at different times.

I’m not enough of a math whiz to figure out your situation, though, and I think it takes a pretty good swing for it to work.