Long Run Probability Question

[Huskiez]

This may have been asked before on this forum; I'm sorry if it has been.

Suppose I have been offered a bet where I am the favorite. My question is, in the long run, if I take that bet an infinite number of times, I will at some point end up broke, regardless of the (finite) bankroll I started with and regardless of how favorable the bet was? I think that is true, I just wanted to check the answer here.

[ironman007]

No, it's only true if you bet your entire bankroll. If you bet a fixed amount and your bankroll is large enough to withstand the deviation of results you will not go broke especially if you are betting with an advantage. For example, you have $1000 and you bet $10 on the roll of dice, if it's a 12 you lose, otherwise you win and you are given even money on the bet, then obviously you could make this bet over and over again and you will not go broke.

[Huskiez]

But no matter what your bankroll is, there's always a chance that you lose x consecutive bets such that you end up with nothing (x increases as you increase your bankroll), and because you're placing this bet infinitely many times eventually that probability (no matter how small it will end up to be) will occur? That is just my reasoning.

In your example the probability of my losing 100 times (thereby busting me) in a row is 1/(36^100), which will happen rarely, but can happen. If I win my first bet, the probability of my busting is 1/(36^101), which will again happen rarely, but can happen (etc).

I think of it like one bet with all of your bankroll, where the chance of success is insanely high (1 - (1/36^100)) in your example, but because there's a chance of failure not equal to zero (here 1/36^100), you're going to end up broke.

[MrBlini]

You'll keep the bankroll with probability very, very near one.

The rate at which the risk goes to zero in this example is a rapid exponential, much faster than the constant rate at which the number of hands goes to infinity. The sum of the risk of ruin over all hands is hardly any more than the risk of ruin during the first 120 hands.

[MicroBob]

as i understand it.....yes, at some point you can hit zero....the following ideas are from a total amateur so please keep that in mind and feel free to disagree.

if you bet an infinite number of times then this should account for ALL possibilities. this would include the extremely unlikely possibility that you will lose 1-billion bets in a row. obviously, the odds of this happening are ridiculously remote, however it needs to be remembered that infinity is pretty damn big.

in fact, there should also be a spot where you lose 999,999,999 hands in a row followed by winning 999,999,999 hands in a row.

it's something like the monkey randomly typing out shakespeare if it were able to type into infinity.

however, i suspect the odds of getting to see this (either the monkey typing shakespeare OR 1-billion losses in a row) would still be extremely slim even if you were able to roll the dice 100x per second (or have the monkey type 100 keys per second) for 1-trillion years.

sooooo....unless you plan on playing for a REALLLLLLLLY long time, you needn't worry about quite so much about busting out as long as you are playing a winning game.

is this the general concept you were trying to get at??

[BruceZ]

sooooo....unless you plan on playing for a REALLLLLLLLY long time, you needn't worry about quite so much about busting out as long as you are playing a winning game.

It's just the opposite. If you go broke, it will happen relatively early. Once you've survived a long time, your bankroll will have increased to the point where the risk of going broke becomes negligible.


if you bet an infinite number of times then this should account for ALL possibilities. this would include the extremely unlikely possibility that you will lose 1-billion bets in a row. obviously, the odds of this happening are ridiculously remote, however it needs to be remembered that infinity is pretty damn big.

It is true that you will eventually have negative swings as large as you please, but this does not mean that you will go broke. What is important is how rapidly your bankroll increases relative to these swings. If you are playing a winning game, you can make your risk of going broke as small as you like with a finite bankroll. See the bankroll formulas in this thread.

[bigpooch]

No, there is always some chance you will NEVER be broke,
depending on the number of bets in the bankroll and the
chance of winning an individual bet. Even if you start out
with only a bankroll of just one bet and your chances of
winning are just above 50%, there is a small chance you will
never reach zero!

Infinity is a very long time, but an edge is an edge!