The gambler's fallacy is the idea that if you take a coin and flip it two times that it is more likely to end up with a heads on the second try if the first try landed on heads.
It's true that the individual instance of flipping the coin is always 50/50, until you introduce the possibility of a group of coin flips. For example.
Flipping a coin two times and ending up in a situation where you get heads twice, is only likely to happen in 1/4 Instances.
The actual chance you have of landing heads after the first flip is not 50/50.
Because first you had to land heads in order for the possibility of two heads in a row ever occurring.
This has the implication of being a meta-tool for people who want to up their chances of guessing correctly who is Mafia and who isn't, based on what they were in previous games. Because there is a different chance of being Mafia twice in a row than there is being Mafia in one game.
Enough of a chance that people would bet money on it.
The issue is you can consider each coin flip as non-isolated simply because you can take the entire batch of coin flips and then claim that in every instance that the coin flips more than one time, you're actually betting on a specific combination.
This simulation demonstrates what is known as the weak law of averages. Simply put, if a chance is 50/50, the expected outcome of bets is that there will be an even distribution of events toward the left and right.
This creates a visible curve, what people are doing when they bet on the opposite is that they will not encounter an instance where they win a bet where their odds are stacked against them in the billions.
And the truth of the matter is, they don't. The reason why is because the distribution of people who win such lotteries is spread out throughout the entire population of people on the planet. If one in a billion people win the lottery, then it would be unexpected for the rest of the people that participated to also win the lottery, and they don't.
If you want to make a simple way to figure all possible solutions use (H+T)^n, where H+T=1. Then the number of heads/tails is H^k*T^(n-k)*(n choose k). But remember each case is isolated
The solution for this ever happening in Mafia is simple though. Just allow people to choose which games they can make public and which ones they don't want to be public.
Ideally for anyone to make use of these methods though, you'd have to know that players total game history, rather than a few examples, with just a few its possible to narrow odds but to no massive degree.
Your best tool, is actually your own game history. How? Well, simple. When you enter a game room, on your left side is a series of names, we will disregard the names, and simply say that "the first slot has been Mafia X times over the course of Y runs on this game mode", as although name order changes each game, the slot is static. The first box, is always the first box.
This is a screenshot that just has me compiling my previous winnings from the coinflips. My original "points" was 50, I did separate runs, and just added the net gain from the first few for that 81.
Right now I have doubled my points and the pattern seems to suggest that I'm going to be earning even more. Better yet, my odds of losing all of my points go down every time I reach a point that I can risk another bet without going into a negative number.
For example 2, 4, 8, 16, 32, 64, 128, after I reach 128 my next 'safeguard' will be at 256, and I'll be able to bet 1/2 On odds that are actually staked up to earn cash on a 1/256 Bet.
Since game odds can be reduced exactly like the coinflips can, it would be safe to start betting on the odds that after 7 Games of not being Mafia in a set-up like Fancy Pants, that the person is now 'due' for their Mafia role, exactly the same as they would be due for a tails flip according to the rule of averages over a long time.
The issue with averages is, with a run of 1,000 1/2 Flips, you would expect that the average result was evenly divided, and for most cases, it actually is. The only thing that I am doing in these simulations is betting that the averages will settle back into their expected place, because anything else would be 'abnormal' for a 1/2 Chance.
Their odds of guessing are smaller because the amount of possible combinations that I could have landed is not just "all heads" and "all tails". Rather they're 2^50, which is a pretty outrageous number.
Remember that there is only one correct guess, therefore their odds of being right about 50 flips is not 1/2.
However, their odds of being correct increase for every part of the sequence that is revealed, for example if I gave them the results for 49 Of the 50 Flips, their chance of being correct is now 50/50, because the possibilities have been sufficiently eliminated.
What I'm doing is just assuming that the universe is blind, in my case, I could flip the coin 49 Times, and I would be able to bet on 1/2 Odds in a scenario where I should be really betting on something extremely unlikely to happen.
Edit:
It wasn't 2x50, it was 2^50. Which is hard for machines to calculate for being so fuckhuge. (Odds of that person guessing right are 1/1125899906842624).
Hey bro. If someone came in after you just flipped that coin 50 times and it came heads each time, but they didn't see it happen are you saying there odds aren't 50/50?
It was more exciting than the first, but the rule seems to be the same. In both cases, if you're able to continue the doubling, you're just more likely to have a net gain.
The only condition that goes against this system is that if your funds are not unlimited, there is and will always be a small chance that you will lose all of your money.
But over a long time, you're basically assuming that you're the casino, and betting on the odds that your patron doesn't hit your jackpot.
My bets are consistent with the maths, my initial bet is 1, and then when it reaches the second bet, I increase the base to 2.
1/2. If I fail, I double the 2, 1/4. If that fails, I double the fraction once again, for 1/8. This is basically equivalent to adding one to your fraction's exponent every time you fail.
The explanation is simple. One times by anything is still exactly one. An exponent of 1^6 = 1, because 1x1x1x1x1x1 still equals 1. But the possibilities were 2, and two times anything else equals a higher number. For every bet, the fraction's exponent is increased by 1.
The Monte Carlo incident was not famous for demonstrating that odds do not change in group bets. It was famous precisely because it demonstrated an astonishing instance of bad luck.
It's only taken me 13 Coin flips to make a net profit of 9. I'm fairly sure that if the pattern continued, I could become very rich. Let's say for instance that I had £1,000 That I could spend on something luxurious that I didn't need.
I could spend that £1,000 And I have a higher likelihood of doubling it than I do of losing it.
Jackpots are rare, it was unfortunate that the gamblers at Monte Carlo were unfortunate enough to hit one...
My method is simple. I bet 1 Until I lose a bet. Then, I double my bet that the coin that lost will win on the next flip. The pattern is expressed in the screenshot.
you are presuming that the events have relevance in a sequence. they do not. a coinflip, taken individually, will always be a 50-50 chance.
sequencing the events gives you the 1/4 probability. taken separately as separate events, you obtain the accurate probability of 1/2 every time you flip the coin. same goes for mafia roles.
not after already flipping heads, though, just from the beginning. that's the mistake they made at monte carlo
your chances of rolling mafia twice, assuming for simplicity there's 1 mafia 1 villager, is 1/4. after rolling mafia once, however, your chances of rolling mafia is 1/2 (1 1 or 1 0, 0 1 and 0 0 are ruled out).
Probability-wise, this is not true for separate Epicmafia games. The random generation of roles, for example, is an entirely separate event.
It's also not true for the coin flip scenario. They are mutually exclusive events, and the first flip does not alter the probability of the second flip. In explaining the fallacy, you have ironically fallen for it.
It only becomes a 1/2 Chance assuming that the first flip is not connected to the second. If we are flipping twice there is not a 1/2 Chance of landing Heads - Heads.
It's explained like this:
Coin is 0.5, the likelihood of a specific outcome is 0.5.
The likelihood of an outcome in two flips is 0.5*0.5 = 0.25.
We know that in reality every action is connected to a previous one, otherwise the concepts of past events would not affect the present.
Meaning that in order for the possibility of 1/2 For the second throw existing, there must have never been a previous flip. It becomes 1/4 The moment it is called "The Second Flip".
It's a fallacy for a reason. After narrowing down the first flip is heads, then you eliminate the options with tails as the first flip, leaving your only options as Heads-Heads/Heads-Tails.
