Coin Paradox Demystified
According to an account of mathematician Kannan Soundararajan attributed to him in a Quanta Magazine article [1], Soundararajan visited a lecture at Stanford by mathematician Tadashi Tokieda, in which Tokieda mentioned a counterintuitive property of coin-tossing. For the sake of simplicity, allow me to term this so called counterintuitive property of coin-tossing The Coin Paradox.
The Coin Paradox
Here’s the cited description of The Coin Paradox copied from the Quanta Magazine article [1]:
”[…] If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses. […]” – Mathematicians Discover Prime Conspiracy, Quanta Magazine (2016/03/13)
Let us introduce a simple terminology for the sake of demystifying The Coin Paradox: H = heads; T = tails, H, T = last toss completing required result. When we write …
Bob = T H T T H H |
… our terminology translates into English saying that Bob needed to toss the coin 6 times until he arrived at the required result of tossing two heads in a row. First came tails, then heads, then tails, tails again, heads, and heads. Note that the last toss completing the required result is underlined.
Now let us introduce some scoring and say that the completing toss has a score of 2, and the toss immediately before the completing toss has a score of 1. Let us further account for the current score by putting it in between round brackets right next to the respective coin-toss:
Alice = T(0) H(1) T(2) |
That’s how it looks like when both Alice and Bob are lucky enough to only need two tosses to arrive at their required result, respectively:
Alice = H(1) T(2) |
Bob = H(1) H(2) |
Paradox Demystified
The Quanta Magazine article further states right after The Coin Paradox description cited above that, “Soundararajan wondered if similarly strange phenomena appear in other contexts.”
As for the paradox: There is none. Let’s see why …
In terms of probability theory, tossing a single coin repeatedly is an independent event. In other words: The history of previous coin-tosses has no influence on future outcomes of coin-tosses. The chances of tossing heads or tails are the same: 1/2 or 50%.
Let’s look at two possible coin-tossing histories for Alice and Bob that perfectly illustrate why Alice has better chances to arrive at the required outcome and why Bob’s chances are worse:
Alice = H(1)H(1) T(2) |
Bob = H(1) T(0) T(0) H(1) T(0) T(0) H(1) H(2) |
Let’s elaborate on the two coin-tossing histories
When Alice and Bob each tossed H on their 1st toss both scored 1. For Alice’s 2nd toss, she either scored a 2 if she tossed T (which she didn’t) or her score remained 1 if she tossed a H (which she did). However, for Bob’s 2nd toss, he either scored a 2 if he tossed H (which he didn’t) or his score went back to 0 if he tossed a T (which he did)!
In other words: Alice’s score couldn’t fall back to 0 while Bob’s score could. That’s where the unequal chances come from. No paradox. Just unfair conditions for Bob.
Takeaway on strange phenomena and paradoxes
There are no strange phenomena and paradoxes outside those produced by suboptimal conceptual frameworks used for interpreting the data perceived by our senses. Change the conceptual framework to make strange phenomena and paradoxes disappear. If there is a sound explanation for one thing, there must be a sound explanation for everything.
If you still think The Coin Paradox is a paradox, you probably fall victim to some form of wrong attribution error.
Footnotes:
[1] https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/