https://dailypost.wordpress.com/prompts/conundrum/

This is a short post that I decided to make because I have been playing a game called ‘Secret Hitler’ by the makers of Cards Against Humanity with my friends recently. It’s a really cool game and I might cover it in a future post but for now I’d like to explain the trouble my friends seem to be having with The Gambler’s Fallacy.

__The Problem__

In the games of Secret Hitler I have been playing with my friends, there was a streak of 5 games where I was a ‘Liberal’ and not a ‘Fascist’ (these are roles in the game). Now my friends were convinced that I was a Fascist in the 5^{th} game with no supporting evidence. The reason was because I had gone 4 games without being a Fascist. I tried to explain that they were wrong but they wouldn’t listen (still won by some miracle).

__What is the Gambler’s Fallacy?__

Firstly, I’d like to clarify that the above situation is mathematically ridiculous. This is not how statistics work. The Gambler’s Fallacy is defined as the common misconception that if you achieve any one outcome more frequently than another, it makes the other outcome more likely as a result. This is not true. Regardless of what result was obtained prior, you are not any more likely to achieve a different result. I would like to point out that The Gambler’s Fallacy only applies to situations where the outcome is truly random.

__An Example of The Gambler’s Fallacy__

Let’s say, for example that you flip an ** Unbiased Coin that has Heads on one side and Tails on the other**. The odds of you getting Heads are 50% and the odds of you getting Tails are also 50%. Let’s say that, in this case, the result of this coin flip was Heads.

Now let’s say that I flipped another Unbiased Coin with Heads on one side and Tails on the other (directly after that one). What is the probability of getting Heads? (If you are interested in taking part in this experiment then do not read the next paragraph before you choose your answer).

The odds of the second coin landing on Heads are:

50%

If you thought it was 50% then congratulations! You understand statistics or were able to guess the result by judging the direction this post was going (in which case you get a cookie). If you thought it was 25% then you made the same common error that most people make. If you didn’t think it was either of these answers… then I am sorry to say that you are not very good at Mathematics.

__Why 50% is Correct__

Regardless of what result the first coin flip produced, there are only 2 possible outcomes for flipping a coin: Heads and Tails and, as we established earlier, the odds of getting either Heads or Tails on an Unbiased Coin with Heads on one side and Tails on the other are 50% both ways.

__Why you may have thought the answer was 25%__

The issue people tend to have is that they consider all the information they have and link it all together. If you got it wrong then I’m willing to bet that you asked yourself what the probability of getting Heads twice in a row is (to which the answer is 25%). But I didn’t ask you what the probability of getting Heads twice in a row were. My exact words were ‘Let’s say that, in this case, the result of this coin flip was Heads. Now let’s say that I flipped another Unbiased Coin with Heads on one side and Tails on the other (directly after that one). What is the probability of getting Heads?’ This question tells you the result of the first coin flip and asks you for the probability of the second coin flip being Heads. Since the first result has been given, you were asked for the probability of getting Heads on a __Singular__ coin flip. Not over the course of two coin flips, just one.

__And just because I know someone will think this__

I know that someone who reads this post will be thinking something along the lines of “But he just admitted that the probability of getting Heads twice in a row is 25%! That somehow disproves everything he just said!” And to that I say:

No. No it doesn’t.

__Why I am Still Right__

So in the scenario presented, the person would be half-right. It is true that the odds of getting Heads twice in a row are 25%. However the odds of getting Heads on the second coin flip are still 50%. This is because, in this scenario, we do not have a guaranteed first result. Now the first result can be both Heads and Tails. Now to make this point as clear as possible, I will make a table showing the percentage possibilities of getting Heads and Tails on the second coin flip using all of the information on hand.

Result of 1^{st} Coin flip |
Percentage chance of result | 2^{nd} Coin Flip (showing combination of results) |
Percentage Chance of Result | Odds of getting Heads or Tails (on second flip) |

HH | 25% | |||

Heads | 50% | HT | 25% | Heads = 50% |

Tails | 50% | TH | 25% | Tails = 50% |

TT | 25% |

Now that it is out there in front of you, can you see it? No? Well I’ll continue then…In this table, we can see that the odds of getting Heads on the first flip are 50% (same with Tails). And then: the odds of getting Heads twice in a row are 25%, the odds of getting Heads and then Tails are 25%, the odds of getting Tails twice in a row are 25% and the odds of getting Tails and then Heads are 25%.

There are two results there that make it possible to obtain a result of Heads on the second coin flip (HH and TH). Then you must add the two probabilities together (25% + 25% = 50%). As you can see, we have a 50% chance of getting Heads on the second coin flip. Similarly, if you were to do the same for Tails you can see that there are two results that end in getting Tails for the second coin flip (HT and TT). Again, we add the probabilities together (25% + 25% = 50%). Again, 50% chance of getting Tails on the second coin flip.

Regardless of how you approach this, if you are trying to find out how likely you are to get Heads (or Tails) on the second coin flip, you should always get a result of 50%.

__How this Relates to Me and Secret Hitler__

In a 10 player game of Secret Hitler there are 4 Fascists and 6 Liberals. This means that there is a 60% of being a Liberal and a 40% chance of being a Fascist. This works in the same way as the coin flip scenario. In one game the odds of being Liberal are 60%. The odds of you being liberal twice in a row are 36%. However, when that possibility is there, the odds of being a Liberal and then a Fascist are 24%, the odds are the same for the opposite (odds of being Fascist then Liberal = 24% and the odds of being Fascist twice in a row = 16%). Now add the two probabilities that involve being a Liberal in the second game together and you get 60 (and when adding the two results that end with being a Fascist in the second game you get 40). Wait… the probabilities of being a Liberal Versus a Fascist were… 60% and 40%!

…Mind blown…

__Final Word__

If any of the friends I played Secret Hitler with are reading this then I would also like to point out that, if you look at the statistics, you are statistically more likely to be a Liberal than a Fascist. So yeah. You also have a 10% chance of being Hitler! A sentence I never thought I would type out in my life…

I hope that anyone who read this enjoyed it! If you want to look smart you can impress your friends (and/or Maths Teacher) by telling them about what The Gambler’s Fallacy is. Or maybe you’ll find a practical use for this… I don’t know.

Either way, thanks for reading!