An Unexpected Probability Result
From Stand-Up Maths (a very short read!)
A surprising fact about probabilities sparked Matt Parker’s curiosity. The random distribution of the square root of X in the range {0, 1} gives the same distribution as the maximum of two random numbers! Let’s simulate that and see how that works:
Let’s compare two simulation runs, with 100 samples vs. 10000 samples per cell:
Matt Parker (the Stand-up Math guy) then tried to find a way to simulate the same thing, with a set of dice. But how? Another surprising result!
A nice little reminder: different mechanisms can hide the same distribution underneath.
Get Pascal Bercker’s stories in your inbox
Join Medium for free to get updates from this writer.
Grant Sanderson (3Blue1Brown) then left a comment adding the following interesting observation:
That was fun! For all you math whizzes, here’s the challenge mode puzzle. Sample three random values uniformly in [0, 1], call them x, y, and z. What distribution describes (xy)^z. Answer: It’s a uniform! This is wild to me, and even though I can prove it formally (and somewhat tediously), it still feels mysterious. If any of you can think of a good way to “see” it, feel free to send it my way and I’ll strongly consider making a video on it.
Let’s see: As usual, I run two samples, with just 100 and then 10000 samples per cell, to see the difference.
In his YouTube video, he left a public link to the ChatGPT discussion that helped spark his curiosity (see link below) and explains why this occurs.
