I Flipped A Coin 3 Times In A Row And It Came Up Heads... What Are The Odds It Comes Up Heads On The Next Flip?

If you said any number other than .50 (or 50%) you just fell victim to the Gambler's Fallacy... Don't worry you are not alone, many people fall victim to this line of thinking. 

What is the Gambler's Fallacy you may ask?

The Gambler's Fallacy is defined as:

The mistaken belief that if something happens more frequently than normal during some period, it will happen less frequently in the future. And inversely, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature). (Wikipedia)

Basically it is where people see certain events happen a bunch of times in a row so they tend to take the opposite side thinking that since such an event has happened so many times already the opposite outcome is "due". That line of thinking has lost people many a dollar at the roulette table...

In fact the Gambler's Falacy also goes by another name because of a famous incident that occurred involving a roulette table at a casino... "The Monte Carlo Fallacy".

Back on August 18, 1913, there occurred one of the most famous examples of the gamblers fallacy ever recorded. At the Monte Carlo Casino, the ball had fallen black 26 times in a row at the roulette table. This was an extremely uncommon occurrence and upon hearing this, gamblers flocked to the game betting massive amounts of money on "red" thinking, incorrectly, that since "black" had come up so many times in a row, "red" was due to come up any time. Needless to say millions of dollars (actually francs) were lost that day as there is no imbalance in nature that must be corrected no matter how many times the ball lands on "black". 

Each event is independent of every other event. The wheel and ball don't know that they landed on "black" the previous spin and therefore there is no imbalance that must be correct... there is no history that can influence future outcomes.

Now what if I told you I flipped a coin 50 times and it came up with heads 50 times in a row... what would you say the probability is of that next coin flip also coming up heads? 

This is where it gets a little tricky... if we can determine that each coin toss really is a mutually exclusive and independent event we could safely say that there is a 50% probability that the next flip will also be heads. However, when we have a long list like this one has to wonder if maybe these events are not equally likely... one has to wonder if perhaps the person flipping the coin has some kind of skill where they can reliably make a coin land on heads... this would be a case where the probability is unknown. 

When the probability is unknown we can safely assume, using the Bayesian Inference, that the previous outcomes demonstrate the likely direction of the bias and the outcome that has occurred the most in the observed data is most likely to continue in the future. 

In our instance... heads.

What is the Bayesian Inference?

The Bayesian Inference is where the probability of an outcome is constantly updated as more evidence or information becomes available. It is an important technique in statistics, and especially mathematical statistics. In our coin toss example, the more times a coin lands on heads the more data and information we have collected. Thus we can safely assume that the coin toss is not an independent event. Either it is not a "fair coin" or there is some skill on the part of the person tossing the coin or some other outside force that is making the outcomes not equally likely. 

Therefore when deciding what the probability is of a certain event, the first thing we must do and be certain of is that the event truly is an independent event with both outcomes being equal. If they are not, then we must use the Bayesian inference and draw our conclusions from the data available. 

In the case of the roulette table... had this principle been followed, people would have bet on "black" instead of "red" and continued to make large amounts of money as "black" continued to come up.

Live well my friends!

Sources:

https://en.wikipedia.org/wiki/Gambler%27s_fallacy

Image Sources:

All images are courtesy of Pixabay.com

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