In his book *The Black Swan*^{1}, Nassim Nicholas Taleb, a fellow urban slow-walker, describes a scenario in which he poses the following question to two characters, the rational & educated Dr. John and the intuitive & streetwise Fat Tony:

*Assume that a coin is fair, i.e., has an equal probability of coming up heads or tails when flipped. I flip it ninety-nine times and get heads each time. What are the odds of my getting tails on my next throw? *^{2}

Dr. John refers to the question as trivial and gives the mathematically correct answer of one half. Fat Tony calls Dr. John a sucker and says,”no more than 1 percent, of course … the coin gotta be loaded.”

This gets at a critical disconnect, noted often by Taleb in *The Black Swan*,* *that arises when we endeavor to generalize a real-world-applicable probability calculus from neatly devised games (application of which he aptly calls the *ludic fallacy*). This distinction between probability models and the real world is one I often struggle with in my ongoing attempts to understand the tense relations between formal probability, intuition (what I sometimes call *informal probability*)^{3}, complexity, and epistemology (i.e., belief, opinion, knowledge). In short, I’m with Fat Tony: If I ever saw someone throw 99 Heads in a row, I’d think the game rigged.

To be clear, Taleb’s example urges us to go further than simply suspecting fowl play should we encounter a real-world instance of 99 Heads in a row. I presume the rational Dr. John would also be skeptical in that situation. What’s questioned in the example, rather, is whether we should accept such a scenario even on conceptual or theoretical grounds. This is what Fat Tony refuses to do by rejecting the thought experiment itself.

I’d like to explore this theme further, starting with a similar question: What is the probability of throwing 100 Heads in a row?

Some thoughts (I used Wolfram|Alpha for the math):

This is an easy question to answer. The probability of flipping a fair coin and getting 100 Heads in a row is 1 in 2^100. That’s 1 in 1,267,650,600,228,229,401,496,703,205,376.

Or, written out: 1 in 1 nonillion 267 octillion 650 septillion 600 sextillion 228 quintillion 229 quadrillion 401 trillion 496 billion 703 million 205 thousand 376

Or, in decimal form: .0000000000000000000000000000007888609052210118054117285652827862296732064351090230047702789306640625

In other words, the probability is very, very, very, very low. Not zero, but might as well be (as far as I’m concerned).

And the probability of getting *at least* one Tails in 100 flips is: 1 – (1/2^100). Continue Reading