By Nina Peloso
As people, our first impression of the ‘chances’ is almost always reliant on ‘intuitive probability‘. Most often, this habit helps us by letting us make broad assumptions that (usually) turn out to be correct. However, sometimes our intuition is what bars us from wrapping our heads around the actual solution.
In 1885, Lewis Caroll’s “Pillow Problem” spurred a flurry of discourse and confusion over whether the solution was “reasonable”, not to mention if it was really correct. Over 100 years later, Marylin Vos Savant’s explanation of the ‘Monty Hall Problem’ flung the mathematical community into another clash with intuitive probability.
Problems of probability are usually not so up to debate. The probability of a coin flip, binomial distribution, and even Bayes’ Theorum are some examples that are seen as unanimously true. It’s the probabilities that seem to ‘not make any sense’ and challenge our intuitions that go on to bemuse and anger the public.
Intuition & The Birthday Paradox
The ‘Birthday Problem’ states that in a room of 23 people, there is just over a 50% chance of two that share a birthday. When the number of people reaches 75, the chances grow to over 99.99%.
At first glance, you might point to the pigeonhole principle and reason that the only way to reach a 99.99+% chance is to gather a room of 367 people. That way, every day (including leap-years) is accounted for.
However, the ‘Birthday Paradox’ has been proven, and is only regarded as a “paradox” because it tricks our intuition.
Often questions of probability threaten our intuitive-leaps because they rely on exponents instead of division, which is much harder to guess.
Let’s take for example, the common coin-flip.
Now, what’s is the probability of flipping ten heads in a row? 5%? 1%?
Relying on the ‘factor tree‘, or division method, would leave us with a 1:20, or a 5% chance. However, through exponents, we can figure out that the chance of getting ten heads in a row is actually closer to 0.001%, or the 1/2 chance of a single-flip, compounded ten times.
The method is simple, but butts-heads with what first comes to mind for the average person. Not dividing the initial probability by the number of times, but decreasing it exponentially.
How Does Coin-Flipping Apply to the Birthday Paradox?
Both questions rely on exponentially increasing odds, and can be solved in a similar way.
Let’s equate the Birthday Paradox to a question of coin-flipping. For example, “What are the chances of getting 1+ heads if you flip a coin 23 times? The best way to solve this is to first invert it, because we know that the chances of all-tails vs. 1+ heads add up to 100%. Using the same formula as before, we know that the chances of getting all-tails are about 0.5^23, and so the chances of 1+ heads are 100 – 0.5^23, or well over 99%.
Well, let’s apply the ‘inversing’ solution to the Birthday Paradox!
First, the chances of any two of these people not sharing a birthday are equal to 364/365 (excluding leap-years), or the days left to choose from after the first person’s birthday is established.
Secondly, we know that in a group of 23 people, there are 23*22/2, or 253 possible matching-pairs. This is because that every one of the 23 people can be paired up with 22 different people. Plus, since half of the pairs are redundant, we divide by two to deduce the number of unique pairs.
From here, we can use the same formula of (initial probability)^the number of rounds on the Birthday Paradox:
(364/365)^253 = about 49.995% or the chances of no-one sharing a birthday.
So, 100 – 49.995 = the probability of one pair that shares a birthday, which is just over a 50% chance!
Because of the exponential growth that occurs once you increase the number of people in the room (and thus, the number of possible pairs). After you get past 70 people in the room, the chances grow to well over 99.9999%.
So in the end, this “paradox” of probability is really just some nearly intuitive-proof combinatorics at work.
If you’d like to try it out yourself, here’s an interactive-simulator of the Birthday Paradox in action!
And a further explanation of the paradox using Python!