Originaly writed in March 2026, in SKG
Disclaimer:This is a one take-brainstorming article ignore anything informal,or grammatical errors.I just want to put my message across.
Disclaimer2:You should definetely pay close attention to what i am writing,any misuderstandings is not my fault.
The ultimate goal of this article, is to explain via examples how insane is playing in unfair games in real worlds scenarios and present you an insane ,yet mathematically sound, paradox. This article might be long ,as i want to implement mathematical proofs to be as rigorous as possible. But first ,we should start by briefly defining some terms we would use. I would use LaTeX wherever math can get complicated, but i can’t do anything for the white color.I’m sorry
Expected value and House edge
Expected value of a game: E[X]=p * (b * (odds-1)+ q * (-b) ,where b: is the bet(money wagered) odds: is the decimal odds as european use. The term (b*(odds-1) is basically the amount you would win if you bet b p,q:= probability of winning ,losing respectively. We can easily see,that in casino games where house edge exist in favour of the house, p<p_real and q>q_real so we can see how E[X] can turn negative.
Probability(of winning): where w:=odds-1 the amount of winning. so Probability of losing: . On the other hand House edge, is the percentage the house expect to win of the total staked money. H.E=E[X]/b = q-p*(Odds-1) , in decimal odds(its better for sport betting), NOTE: be carefully bigger odds don’t necessarily means bigger H.E ,as q=q(odds) and so is p,so the probability changes in different games. In fractional odds we have H.E=q-p * (odds)
The insane paradox of “All in” betting.
We have all heard that going “All in” in a casino game is a braindead,degenerate move …but is it? Lets say we have the average gambler that want to make his money k times more He is thinking of 3 strategies : Strategy 1(Y_1): Betting his whole bankroll in a single bet. Strategy 2(Y_2): Dividing his bankroll and betting each time Bankroll/N units , so that Bankroll modN=0 and N>1 ,for N bets(So N:=the number of bets). Strategy 3 (Y_3):The third strategy similar to second but,he stops untill he hit the goal or ruin.
(let k=2 cause our analysis would be really stohastical and mathematically heavy, and we are betting in a game of odds 2.. lets say black or red in roullete…The stohastic approach to k,and odds as a parameter is really difficult to understand so lets go by a real casino game….however we can see that k,and odds doesn’t really matter.) Now,we would find E[Y_1],E[Y_2], P(reaching goal | Y_1),P(reaching goal | Y_2),P(ruin|Y_1) and P(ruin|Y_2), and so for the third.
Mathematical analysis of strategy 1 vs strategy 2.(Variance)
Someone would look this and say: , and
So E[Y_1]=E[Y_2]. So it doesnt matter we are gonna lose the same. But there is something else called variance.The bigger the variance the more chaotic a random variable (such as profit) could get ,here is where luck lives.If you play an unfair game, Negative expected value game, you want to maximize chaos. You want to get lucky before that EV tear you down(Law of Large Numbers). So what if we manage to maintain high variance?what is the probabilities for each strategies?
(Remember P(A | B)= P( of event A given(in case of) B) .
we can see that so for most games somewhat 48%. .
we can easily take the limits as N approaches infinities and since (1-h.e/2) <1 and so is (1+h.e)/2<1 ,we have P(goal | Y-1)->0 so practically 0!. so by following this strategy the probability of you hitting that goal is virtually impossible and it gets lower for more games you are thinking of playing, but so is the probability of you losing the whole bankroll!).If you like the thrill of gambling this strategy this strategy means you will play more and just enjoy the night in the casino, but you are mathematically guaranteed to fail.
Strategy 3:The gambler’s ruin paradox
Now instead of N bets, you bet till you hit that goal or loss everything. You can find the derivation of the formula ,but this would take several lines to write:
lets take the limit as N approaches infinity we can see that: P(goal | Y_3)=0.(why? the denominator get bigger as n get bigger,1+H.e>1-H.e) and
and its limit as n approaches infinity is 1.
A numerical Example
So lets say our bankroll is 1000 units and we want to make it 2000 in (we choose N=100 so we bet 10 units per round.) roullete and lets say we bet on red(red or black it doesn’t matter) =48.6%it doesn’t rely on N. =0.44% and to find P(ruin | Y_1)=1-P(goal | Y_1)=51.6% or P(ruin |Y_2)=1-0.44=99.56% Imagine now if N=10000 (!)
Thats the paradox! I would say its not a paradox rather it proves why playing in negative e.v is so dumb that even absurd thing like that sound normal! So what we should do?Bet all our bankroll in red or black?Hell no,that’s financial suicide.Avoid betting in negative E.V in general!! But we see something intresting,If you have to play in negative E.V you should maximize variancee,meaning you should do risky strategies,and rely purely on luck,or else the law of large numbers would put you in a bad position.
“All in” in sport Betting
Now let’s step out of the casino and into sports betting. Let’s assume you read my last articles, you updated your Bayesian priors, and you actually found a sharp edge. You are now playing a +EV game. Should you go All-In now? Absolutely not. It is suicide. We should think what should we bet…and the total stake we choose is as importand as finding a value bet. We cant simply “all in” cause P(whole bankroll ruin) is very high due to short-term variance. Nor can we play really small bets,or we won’t make a good total profit. Here is where kelly criterion comes in! Something we gonna talk in the next article eventually ,and then start applying everything we learned !
Final thoughts
For some gambling is about a hobby ,for others a way to make money…however we can all agree that for some is addictive and catastrophic.Either way be responsible, be mathematically correct to minimize losses(you see i dont say win -because that is not always the case) and enjoy! And by writing this i just thought of something;casino tendancies to achieve low variance,and low output potential via house edge is the way to steal not only your money ,but also your TIME…some american would say that’s the two basics in today lives.
KING KOTSIKAS,SKG, MARCH 2026