Aaaa'right, here I come again. As someone said, all that mathshit obfuscates the reality of what happened.
It will be slightly over too much, certainly off-topic, but I am only writing this
partial analysis for entertainment, for personal interest and for educational purposes
May contain errors.
The eventMateonl robbed the bank.
The question (?)How lucky was he ? Very. That's a non mathematical question, so that's a non mathematical answer.
To push the analysis a little bit further, one needs to consider two (ideally mutually exclusive) answers
a) he was just lucky,
b) he was way too lucky, game was rigged.
"Methodology"To decide between a) and b), we need to
- ask a mathematical question "What is the probability P that... "
- fix a limit number P0 such that if P>P0, well conclude that it was luck, if P<=P0, we'll conclude that it was rigged
- compute P, decide.
Ain'no Joe Got' Time fo Dat!
Before formulating zee questions and the before zee Germans arrive, I spy with my little eye
Three Approaches to answer a "what is the probability..." question
1) statistical study based on observations
2) choosing a model and making a statistical study based on simulations
3) choosing a model and perform a mathematical analysis (purely theoretical approach)
Depending on the problem and the amount of information and knowledge available, each approach has its own benefits and drawbacks. In a perfect world, if we do a good job, 1) and 2) and 3) should be all consistent.
About approach 1, since we've seen only one mateonl guy rob the bank only one time, our sample size is two or at best two (Nakowa's case, should they be comparable). From a purely observational point of view, no conclusions can be made with only one observation other than "the observations show that if you bet like manlteo and have the same initial fund, you WILL rob the bank". Well, good luck with that!
Do we all appreciate the elegant Simplicity of simulations and the strange Beauty of statistical theory. Yes ? No ?!
Doesn't matter, I'll do none of'em anymore.
Assumptions, available information and the probability questionsOk we're choosing a model. Ideally, it should be as close as possible to the reality. All bets are @49.5%. That's our model, but also the reality. Lucky us.
Mateonl has a finite initial fund. So he can bust. And our probability will depend on that parameter which is.... tuduuu unknown. At all time, bankroll cannot go in the negative. So bank can bust too. Recall that, obviously, mateo didn't bust. Indeed, if he did, he would have stopped on a loss, which is not what happened. From a probabilistic point of view, these two statements must be taken into account, I don't repeat them below.
For the sake of simplicity, many approximations are to be made, among which : now new investments in the bankroll and no withdrawals were made during the bets, no other whale neither etc etc. I also wrote numbers out my memory, just for illustration.
The questionsI will formulate two seemingly similar questions.
I)
Given this list of bets which are all realized (in this order)
5k bets at 7
5k bets at 3
25k bets at 1
25k bets at 0.5
what is the probability to end up wining 85% or more* of the initial bankroll ?
*(if we are in the mindset of thinking "what is the probability to be
at least as lucky as mantoel")
II)
Given this list of bets which are realized (in this order) so long as the bankroll contains more or just 15% of the initial bankroll:
at most 5k bets at 7
then at most 5k bets at 3
then at most 25k bets at 1
then at most 25k bets at 0.5
what is the probability to end up wining 85% of the initial bankroll ?One can arguably say for question I) that it allows the bankroll to go under 15% of its initial state, which is not exactly what we saw happening. But again, if we think of the probability "betting like maeto did, to be luckier than him" as "to win more than 85% of iBR " it is a reasonnable assumption.
Similarly for question II) it allows the betting to finish before 60k bets, which is not exactly what we saw happening. But again, if we think of the probability "betting like maeto did, to be luckier than him" as "to win .xx. btc quicker than him " it is a reasonnable assumption.
Mmh strange.. none of those replicates what we saw!!! Let's take the two conditions into account: what is the probability to make exactly 60k bets like he did, win exactly 85% BR, BR never go under 15% before the end. Given the large number of bets, I think that modelling this would involve a massive shitload of combinatorial arguments (partitioning of numbers). Approximation would probably be the way to go... but let's stick with I) and/or II) and... ?
Now about dooglus'
simulation, the probability he finds is the answer (I think) to this question :
III)
Given that
- all bets are continuously updated at the maximum bet size (which is 0.5% of bankroll) all the time at most 60k times,
- betting stops if bankroll reaches 15% of initial BR,
what is the probability to win 85% of the initial bankroll ?This formulation does not make use of the available information about matnl bets. So it'd be of interest to answer this question: betting at max bet at most 60k times, what is the prob to be as lucky as him (in other words, to win exactly what he won).
This last question is of interest essentialy for the casino only as it will help determine the probability of ruin or survival time to a whale (who'd bet only @49.5% .. mmh).
This is just three out of a quazillion choices. But they each help to answer slightly different questions.
You might wonder "what is the probability to end up wining 85% (or 100%) of the initial bankroll ?" without any restriction on the betting strategy. Well it's a Gogol-fucking-infinitely complex question as it involves all allowed betting strategies. Remember your math course ?
P(win 85%) = P(win 85%|Strategy1)P(Strategy1) + ... +P(win 85%|Strategy-BIG)P(StrategyBIG).
I'm done here...
I did some biiiig simplifications on my previous posts and estimates, where I choose the analytical approach. Its power is limited with more complex models..
, and you arrive at something like 45k bets.