Suppose infinitely many blindfolded people, including yourself, are uniformly randomly arranged on positions one meter apart numbered 1, 2, 3, 4, ….
Intuition: The probability that you’re on an even-numbered position is 1/2 and that you’re on a position divisible by four is 1/4.
But then, while asleep, the people are rearranged according to the following rule. The people on each even-numbered position 2n are moved to position 4n. The people on the odd numbered positions are then shifted leftward as needed to fill up the positions not divisible by 4. Thus, we have the following movements:
1 → 1
2 → 4
3 → 2
4 → 8
5 → 3
6 → 12
7 → 5
8 → 16
9 → 6
and so on.
If the initial intuition was correct, then the probability that now you’re on a position that’s divisible by four is 1/2, since you’re now on a position divisible by four if and only if initially you were on a position divisible by two. Thus it seems that now people are no longer uniformly randomly arranged, since for a uniform arrangement you’d expect your probability of being in a position divisible by four to be 1/4.
This shows an interesting difference between shuffling a finite and an infinite deck of cards. If you shuffle a finite deck of cards that’s already uniformly distributed, it remains uniformly distributed no matter what algorithm you use to shuffle it, as long as you do so in a content-agnostic way (i.e., you don’t look at the faces of the cards). But if you shuffle an infinite deck of distinct cards that’s uniformly distributed in a content-agnostic way, you can destroy the uniform distribution, for instance by doubling the probability that a specific card is in a position divisible by four.
I am inclined to take this as evidence that the whole concept of a “uniformly shuffled” infinite deck of cards is confused.
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Question for ChatGPT:
"How and why exactly and precisely is an infinite discrete uniform distribution impossible?"
ChtGPT's response:
"An infinite discrete uniform distribution is impossible because of fundamental constraints in probability theory. Let's explore exactly and precisely how and why this is so.
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🔁 What Is a Discrete Uniform Distribution?
A discrete uniform distribution over a set S means:
- Each element x∈S has an equal probability.
- The sum of probabilities over all elements in S must be exactly 1, as required by the axioms of probability.
💡 The Issue with Infinite Sets
Suppose we try to define a uniform distribution over an infinite countable set like:
S = {1, 2, 3, 4, ...}
Let’s say we assign each element a constant probability p. Then the total probability is:
Σ(n=1→∞)(p) = p+p+p+... = ∞·p
Now consider possible values for p:
Case 1: p = 0
Then:
Σ(n=1→∞)(0) = 0
But this violates the requirement that the total probability must be 1.
Case 2: p > 0
Then:
Σ(n=1→∞)(p) = ∞
Which also violates the requirement that the total probability must be 1.
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🧠 Why This Happens (Deep Reasoning)
The root problem is that uniformity demands equal weight, but probability demands the sum of weights = 1.
- In finite sets, this is fine: you can assign 1/n to each of n elements.
- In infinite sets, you can’t assign any nonzero equal weight that adds up to 1.
- You also can’t assign zero weight, because then the total is zero.
This contradiction makes a discrete uniform distribution over an infinite set impossible.
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⚖️ Continuous Uniform Distributions Are OK
There is a uniform distribution over the continuous interval [0, 1], with a probability density function (pdf) rather than probabilities assigned to points:
f(x) = 1 for x∈[0, 1]
Here we use integrals, and the total area under the curve is 1, so it’s allowed.
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🚫 Summary of Why It’s Impossible
A discrete uniform distribution over an infinite set like ℕ is impossible because:
- You must assign the same nonzero probability to each outcome → leads to an infinite total probability.
- You can’t assign zero probability to each outcome → the total remains 0.
- There’s no way to make the sum of infinitely many equal numbers (except 0) add up to exactly 1.
Therefore, an infinite discrete uniform distribution violates the axioms of probability theory and cannot exist."
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