Probability of getting heads x times in a row decreases rapidly with increasing x.
Probability of getting heads with each individual toss of the coin, however, does not change at all, regardless of how many time it has been tossed.

These are two different things, and you can't rely on either. If all you do is to wait for the opposite to happen, you will end up winning rounghly 50% of the time - which is not accurate either as true randomness doesn't really like equal division. The coin has no memory of what happened during the last toss. The one operating it has it tho, and I would not be surprised if videogames in particular were rigged towards particular results to reduce randomness.

Don't worry, it's not just you -- casinos in fact are able to do so well because it's not a notion that even highly intelligent individuals can easily come to terms with even after they've been shown it, and it is called the Gambler's Fallacy for a reason. Gamblers on a losing streak will have this fallacy in mind when they think that they're owed something or "due" on their next roll of the die, slot pull, or hand and hence, continue gambling even when they continually lose and their odds are exactly the same as they were the last time (stacked in favor of the house).

Your example is actually a good way to illustrate the explanation.

It is true that

XXXXXXXXXXXXXXXXXXXXX is almost impossible to get for a result of 21 flips of a fair coin.

Assuming a fair coin, the odds of flipping 20 heads in a row and then one more head for toss #21 is indeed, extremely low at 0.5^20 (20 heads in a row) x 0.5 (50% chance for toss #21 to be a head) = 1 in 2,097,152.

However,

XXXXXXXXXXXXXXXXXXXXY is equally impossible to get for a result of 21 flips of a fair coin.

The odds of flipping 20 heads in a row and then one tail for toss #21 is also extremely low, and in fact, the same as the above situation at 0.5^20 (20 heads in a row) x 0.5 (50% chance for toss #21 to be a tail) = 1 in 2,097,152.

And so we can see that the 21st coin toss still has a 50/50 chance of heads or tails, independent of any of the results that preceded it.

That said, Fenixp does make a good point that the video game you are playing may not be using a fair coin and could be adjusting results based on an algorithm which you have cracked and exploited to your benefit.

This reminds me when a friend of mine at school wouldn't believe that winning lottery with numbers 1, 2, 3, 4, 5, 6, 7 (seven numbers right) is just as probable as with any other set of seven lottery numbers. His only comeback was "How many times have you seen lottery ending with those exact numbers? Like never.".

I had to point out to him that it is still just as probable as getting any other specific set of seven numbers.

(It is not such a good idea to use those exact numbers though, as in reality there are allegedly quite many others who already use them. So if you hit the jackpot with numbers 1-7, you'd have to share the winnings with quite many other winners.)

The funniest thing is that later in his life he got his Master's Degree from the university of economics. and is nowadays working in the banking sector, trying to estimate and track fluctuations of different currencies. I bet he knows quite a bit more about probabilities and such nowadays, much more than me. Or at least I seriously hope so. :)

Smart calculations there.

Yep I tested it out again, I was wrong.

EDIT: OK. I see you finally got it.

Closer to 47/47 (the other 6 is house advantage)

Hence the "almost".

I think the game "21" is based on method as far as I can tell.

You mean blackjack? No it has a 3% house advantage, a good portion being if you bust you lose your money, but if the dealer busts afterwards you don't get a refund...

Tested it out, didn't work.

I see the advantage to the house in 21 now.

EDIT: OK. I see you finally got it.

The one exception is when a player is able to keep a card count and changes their bet size according to whether the remaining deck is rich with aces and card values of 10. In statistical trials, it has been shown that 10's and aces favor the player while cards 2-6 favor the dealer. One of the main reasons for the imbalance is because a natural blackjack pays out 3:2 in the player's favor.

Depending on the table rules and number of decks, a skilled player can gain an advantage of maybe 1-3% on the house. Counting cards using only your own brain is not illegal, however casinos obviously do not like it if they detect someone doing it. Avoiding getting banned and/or harassed by the casino is the key, and the part that requires a lot of creativity.

I actually had a big post on my Bloodbowl coin toss theory and how it seems to make perfect sense, and then someone mentioned Roulette... we thought we might have found a fountain of money and quickly deleted all mention of it.

I'm afraid ZFR and rtcvb32 have gone straight to the Casino to gamble their life savings away. Lol.

Seems Gambler's Fallacy happens all the time, it's one of the oddest things I've encountered. Wkipedia says;

"The gambler's fallacy is a deep-seated cognitive bias and therefore very difficult to eliminate."

It's even affected male and female births, fooling people that it's likely to have one gender if more of the opposite gender have been born. There's algorithms by really smart people that say this doesn't work.

This is also one of the oddest threads I've ever been in.

How do you figure? I've just read and done research on some casino games. Roulette is one of the easier ones to figure out the house advantage (2/38 = 5.3%). Blackjack & Craps take a bit more brute forcing to get the math and results. Curiously it's the side bets that give the house the highest advantage, jumping from single to several times higher easily, called usually sucker bets.

But i'm not really going to get too serious on this or in depth, there's plenty of sources you can refer to.

Hmmm. Maybe. You can probably successfully card count on a deck of 52, but many casinos employ 4 or 6 decks (300+ cards). Also their shuffling method usually ends up encouraging what's known as card clumping. Those two things together makes card counting fairly difficult if not impossible to employ. You're better off with a good memory, then see how they shuffle cards in groups and adjusting your memory to assume generally where cards are at. But unless you're really determined or a savant, i wouldn't expect much for results.

Just a joke really, I hoped you guys would be smart enough do some research on the subject before running wildly to the Casino to put all your cash on black, you obviously did, like me. Then I noticed algorithms saying why this coin toss theory doesn't work and how many times it's been tried on roulette.

I'm more worried about ZFR now I see your reply. :p