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ben_
04-14-2005, 01:13 PM
An interesting question i stubbled apon today.

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.