Calculating Real Odds

This is based on a question from Black Swan by Nassim Taleb.

Imagine the following real event:  You are at a vacation resort that will occasionally serve meals with assigned seating where you meet other people at the resort.  You find yourself at dinner with four people you have never met before: Alice, Bob, Carlos, and Darla.  Darla pulls out a coin and says,”I have a 50-50, heads-tails coin, so I thought we’d pass the time by doing some statistics.” She flips the coin and it lands heads.  She flips it again and it lands heads.  She flips it 99 times and it lands heads every time.  “Wow,” she says, “What do you think it will be the 100th time?”

Alice says, “It’s been heads so many times, the odds are overwhelmingly in favor of tails this time.”

Bob says, “The odds are still 50-50, so there is still equal probability of a heads or a tails occurring.”

Carlos says, “It’s been heads so many times, I think it will be heads again.”

With whom do you agree?

The correct answer is Carlos.  It will undoubtedly be heads again.

If you guessed Alice, this is a common case of the gambler’s fallacy.  The odds of a new result, if randomly generated, does not depend on previous results.

If you guessed Bob, your error is more subtle.  You probably ignored the first line of the question:  Imagine the following real event.  You probably assumed this was just a fake, statistics question like you would find in a textbook.  But no, the question specifically asked you to imagine you were really there.  The probability of 99 heads in a row for a 50-50 coin is (approximately) 1 chance in 630,000,000,000,000,000,000,000,000,000.  It is just not realistic for this to actually happen in real life.  Obviously, Darla was lying to you.  It is a trick coin that always comes up heads.

This is not a trick question.  The odds of 99 heads in a row is ridiculously unlikely for a fair coin.  You have never met this Darla person before.  If you were really there, you would never believe the coin was a fair one.  You would think she was putting you on.

This thought experiment is crucially important when considering the case of evaluating risk in the real world.  Calculating statistics can be hard and subtle.  It can take a long time to learn to do well.  However, real risk and uncertainty in the real world often come from things that dwarf anything you can calculate: not knowing what the possible outcomes are, not knowing the probabilities of the outcomes, or worse, thinking you know the probabilities, but are mistaken.  Sometimes, you can be more wrong, by even thinking you can calculate the risk.

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