TL:DR; It's 1/10000 but in a the odds in a given day are very different.
Recently I stumbled upon a Bitcoin betting site called freebitco.in. What's great about this website ( not a shill ) is that this web doesn't charge you anything in order to get started, and instead for every hour you're entitled to a free BTC roll provided that ur not a robot ( although the prize decreases the more you roll ).
One thing that attracts me to the website is that all of its features and games are provably fair and unbiased. That being said, a question sorted of popped up in my mind: "What is the probability that I'll get that sweet 0.015 BTC jackpot given that I only have at most 24 attempts every day to earn it ?"
It was straightforward and easy to come up with a quick solution to the problem. You pick a number out of 10000 possible combinations, and the final probability should be 1/10000.
" Gee, it's just as simple as rolling a dice or pulling one card out of a deck. What's going on in your mind ?"
While the above solution is technically correct, this figure assumes that we hunch over our phones and roll forever ( providing that we neither age nor die ) without anything stopping us from rolling ( aka. long-run frequency). As good as freebitco is , we can't expect ourselves playing continuously for many months and years without getting nauseated at the idea of doing the same thing again and again. You would wonder : "What are the odds of winning jackpots in any given day - be it tomorrow, the next day, or your birthday ?"
Prolly 1/10000 isn't the best answer right ? There's gotta be something more to this answer and we're just getting started.
Firstly, there are a ton of ways to take a sample of 24 random numbers out of 10000. Let's call this simply as 10000 choose 24 (1).
Secondly, it is given that the success rate of winning at least one jackpot with 24 rolls is approx. 0.24% in total, assuming that those 24 numbers are independent and different from each other ( 2 ).
" Ok, Dennis, what's next ? "
Well, folks, this question could be answered by using binomial distribution and plugging (1) and (2) into the formula we get:
If you were to dutifully log into the site and roll once every hour for 100 days straight, you would have 7 or 8 days of winning $150+ worth of Bitcoin. Sure, this does not look like a realistic scenario since we can't afford to wake ourselves up every hour late at night just to push the button. We might at most roll 8 - 12 times a day without getting sick of it, but the probability from here takes a nosedive:
The probability just went down from 8% to just 0.01%-0.286%. In other words, you'll have to roll at least 8 times for nearly 27 years and a half ( or 12 times for 350 days ) in order to win. Seems a bit anti-climactic right ?
In conclusion, if you have a bot that somehow bypasses hCAPTCHA tests and play for you 24/24 you would certainly make a career out of it. But if you intend to roll free BTCs for fun and nothing else, then you shouldn't expect big gains from it, despite the promise the site is trying to sell you.
Inspiration
I got this inspiration to write this not long after I watched 3Blue1Brown's video on binomial distributions.
Limitations ( possibly a disclaimer )
I wish I could do a better job in not giving you false information, but the results brought to you are not without limitations.
As I run through the calculations I made random and sporadic assumptions for the sake of simplification. The boldest one is that no matter how many jackpots you win after 24 rolls, the success rate ( 0.24%) is unchanged because I see jackpot streaks of any kind as a win (winning once is the same as 1 jackpot streak).
Next, the whole calculation only works in a single day. I'm not really sure if playing continuously for a calendar month ( 24 * 30 = 720 attempts ) increases your chance of winning, but when the amount of attempts increase the combinations (the n choose k thingy) seem to blow out of proportion. While the binomial distribution formula does not handle large numbers well, by reducing the proportion of the success rate ( from 720/10000 to 9/125 ) things will be a wee bit more manageable.
Finally, feel free to comment down below if you see any part that should be edited or remedied. Any feedback is welcome !
Nice post