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ben_
04-14-2005, 01:13 PM
An interesting question i stubbled apon today.
All thoughts and comments welcome.

Your bank balance stands at 100.00. You may place up to this amount on one of the following bets. You can stop betting at any time.
You can place a maximum of 20 bets.

The four betting options are as follows:

1)A 100% chance of winning 1.04 times your stake.
2)A 50% chance of winning 2 times your stake.
3)A 33% chance of winning 3 times your stake.
4)A 10% chance of winning 20 times your stake.

So statistically, which is the best route to take to gain the most money?

Cheers

Paul2432
04-14-2005, 02:12 PM
Can you change the amount you bet? If so, I take option 2 and bet 25% of my balance each time.

Paul

ben_
04-14-2005, 02:19 PM
yes you can bet any amount.

Siegmund
04-14-2005, 06:57 PM
We need a bit of clarification on the problem statement. Two questions for you. First,

[ QUOTE ]

1)A 100% chance of winning 1.04 times your stake.
2)A 50% chance of winning 2 times your stake.
3)A 33% chance of winning 3 times your stake.
4)A 10% chance of winning 20 times your stake.


[/ QUOTE ]

1) seems to mean "turn $100 into $104 for a sure $4 profit", that is, 1.04 for 1, not 1.04 to 1.

Does 2) mean "50% of the time I lose $100, 50% of the time I get $200", or "Pay $100 for a 50% chance of getting $200"?
Similarly for 3 and 4.

Secondly, this has several interpretations:
[ QUOTE ]

So statistically, which is the best route to take to gain the most money?


[/ QUOTE ]

Largest expected value at the end? Largest expected value with a fixed probability of going bankrupt? Best chance of reaching a set target? The answer to each is different.

Under the first interpretation of your wagers above, each proposition has the following EV and SD per $100 wagered:

1) EV$4, SD$0
2) EV$50, SD$150
3) EV$32, SD$188.09
4) EV$110, SD$630

Type 3 wagers are always wrong. The choice between 1, 2, and 4 depends on the level of risk you're willing to accept in exchange for an increased chance of winning.

For simple maximum expectation, everything on 4 every time -- but 1 chance in 10^20 of having 20^20 dollars is not most people's cup of tea.

The Kelly system says to bet 5.5% of your roll on #4 at every turn (the Kelly-sized bet on #2, 25% of your roll each turn, gives a better rate of return than #1, but not as good as #4).

Moonsugar
04-15-2005, 12:43 AM
interesting problem i couldn't decide which to put first as my picks of first and second i could easily have switched

but: w/o making some math formula to figure it out i think the best solution is probably to bet ~5% on option 4 repeatedly.

second best is probably to bet 100% on option 1 repeatedly

options 2 and 3 are very close to each other and worse than option 1

hope im close /images/graemlins/smile.gif

kiddj
04-15-2005, 08:17 AM
I'm not a big "risk" person. I'll take option A and bet evrything 20 times. End result: (1.04^20)*100 = 219.11, 119.11 profit.

Paul2432
04-15-2005, 09:06 AM
[ QUOTE ]
The Kelly system says to bet 5.5% of your roll on #4 at every turn (the Kelly-sized bet on #2, 25% of your roll each turn, gives a better rate of return than #1, but not as good as #4).

[/ QUOTE ]

#4 will have higher ROI, but will make less money because the wagers are much smaller. For instance, on the first trial, EV of #2 is $12.50, while EV of #4 is $6.05.

That is why I said take #2 and bet 25% in my OP.

Paul

Siegmund
04-15-2005, 04:32 PM
Hehe... slippery fingers on the calculator strike again.
Yes, #2 at 25% is the better Kelly bet.

sfwusc
04-15-2005, 11:01 PM
I think I would do $25 on #2 20 times. Seems like the best from simple math...assuming lower risk.

all on 4 every time would get you the most EV,,,,but you need a lot of good variance.

SFWUSC

elitegimp
04-16-2005, 02:24 AM
[ QUOTE ]
I think I would do $25 on #2 20 times. Seems like the best from simple math...assuming lower risk.

all on 4 every time would get you the most EV,,,,but you need a lot of good variance.

SFWUSC

[/ QUOTE ]

except when you calculate the EV of this bet, you need to account for the (66%) chance you lose all your money on the first four bets

Kristian
04-18-2005, 06:04 AM
Could you explain your motivation for betting on 2 (or 3)? I don't see any reason for these bets as there is no EV+, just risk.
The correct betting strategy seems to be a combination of 1 and 4 depending on your risk aversion.

Siegmund
04-18-2005, 06:47 AM
As I said in my first post, it's not entirely clear what the OP meant.

If he means paying $1 for a 50-50 chance at winning $2, then yes, that's risk with no +EV.

I and several other responders decided to set that interpretation aside and answer based on the assumption that bet#2 meant losing $1 half the time and winning $2 half the time.

bet#3 is a bad idea under either interpretation. bet#4 is affected only slightly by which interpretation you use.