Being a web-dev in casino makes Me sometimes think on probabilities. Sometimes
I find myself entertaining “win schemes” in my head. There are no such schemes
obviously, but at the start of this article, I had no “math” at hand to break
My “win schemes”, or explain to a “child” why it won’t work. Had to read about
50 pages of the “Е.С.Вентцель Теория
Вероятностей”,
to connect the dots.
TLDR: Read read the wiki on gambler’s fallacy.
Alright the mighty “win scheme”:
The probability of events happening in a row drops exponentially(actually correct claim), when it is enough “tails” in a row, We bet on “heads”.
This “win scheme” has special chapter in gamblers fallacy wiki page, called Monte Carlo Casino, where “black” was the outcome on one roulette table for 26 times in a row.
Keeping it more simple with the coin, Wiki actually explains why “heads” won’t come after 2 “tails”, but such explanations are the reason people gamble for wrong reasons.
So a bit of simple math: individual probabilities multiply,
when we care that some outcome happens a certain number of times.
I make a correct claim that 3 “tails” is to rare to occur. And immediately develop
a get rich quick scheme:
I will bet on “heads” every time I see 2 “tails” in a row.
The rare to occur sequence:
“tails”-“tails”-“tails” = \(1/2 * 1/2 * 1/2\) or \(1/2^3 = 0.125\)
Wow, out of 100 games “tails” will happen after 2 “tails” only 12.5 times, so I
am rich. (Meaning that 87 times it will be “heads”)
I cannot be the only one who at least once thought way, right? Just
to make sure, will also write how my “winning sequence” would look like:
“tails”-“tails”-“heads” = \(1/2^3 = 0.125\)
Yes probabilities are the same. Any predicted outcome has same probability as consequtive “tails” outcomes in a row. Hence when there is a “streak” of “tails” or “black”, the probability of next outcome, should be counted together with whole sequence. Or just as simple as single outcome probability, which is counter intuitive. But these 2 perfectly coexist together, probability of whole sequence and singular event. Whole streak’s probability “settles” the case for “reality needs to level out the distributions”. So as soon as You think reality should even out something, remember to calculate from the start. There is no argument that reality will make the distribution 50/50, there is just no way to know how exactly it will do it.
And a joke before going to simulation results. “A blonde” is asked what is the probability of Her meeting the dinosaur on the streets - \(1/2\) She replied, it is either I meet it or not.
For a simulation I wanted to see
With a help of Math.floor(Math.random()*2), in a loop (1000 times), gotten some data:
| (index) | simulation heads | simulation tails | simulation custom | theory |
|---|---|---|---|---|
| 1 in a row | 508 | 492 | 492 | 500 |
| 2 in a row | 170 | 166 | 160 | 250 |
| 3 in a row | 73 | 69 | 72 | 125 |
| 4 in a row | 36 | 32 | 34 | 62.5 |
| 5 in a row | 19 | 19 | 13 | 31.25 |
| 6 in a row | 10 | 11 | 7 | 15.62 |
| 7 in a row | 8 | 2 | 3 | 7.81 |
| 8 in a row | 4 | 1 | 1 | 3.90 |
| 9 in a row | 2 | 0 | 1 | 1.95 |
| 10 in a row | 1 | 0 | 0 | 0.97 |
Code for simulation should be seen here. Answering original questions:
Number of sequences being below mathematical lead Me to double check with a “truly” random sequence from www.random.org Here is the table from there:
| (index) | simulation heads | simulation tails | simulation custom | theory |
|---|---|---|---|---|
| 1 in a row | 473 | 527 | 527 | 500 |
| 2 in a row | 157 | 179 | 169 | 250 |
| 3 in a row | 60 | 82 | 71 | 125 |
| 4 in a row | 27 | 38 | 22 | 62.5 |
| 5 in a row | 9 | 20 | 12 | 31.25 |
| 6 in a row | 3 | 6 | 10 | 15.62 |
| 7 in a row | 1 | 3 | 8 | 7.81 |
| 8 in a row | 1 | 1 | 5 | 3.90 |
| 9 in a row | 1 | 1 | 1 | 1.95 |
| 10 in a row | 0 | 0 | 1 | 0.97 |
Was expecting values to be close to theoretical ones. Hope I have not stopped reading the book too early to miss explanation of “non-linear” connection between probability and rate of occurance. But it started to get too mathematical, and boring.
Get-rich-quick-scheme failed again.