In his book The Black Swan1, 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