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.
2 comments:
"Infinite Discrete Uniform" by MathHolt
Question to Gemini:
"How and why exactly and precisely is an infinite discrete uniform distribution impossible?"
Gemini's response:
"An infinite discrete uniform distribution is impossible because it violates a fundamental axiom of probability theory: the sum of probabilities of all possible outcomes in a sample space must equal 1.
Here's a precise breakdown of why:
1. Discrete Uniform Distribution Definition:
- A discrete probability distribution assigns probabilities to a finite or countably infinite set of distinct outcomes.
- A uniform discrete distribution means that every possible outcome has the exact same probability of occurring.
- If there are n distinct outcomes, the probability of each outcome is 1/n.
2. The Problem with Infinite Outcomes:
Let's imagine we have an infinite discrete sample space, say the set of natural numbers ℕ = {1, 2, 3, ...}. If we were to assume a uniform distribution over this infinite set, let ℙ(X=k) be the probability of any specific natural number k being chosen.
We have two cases for the value of ℙ(X=k):
- Case 1: ℙ(X=k) = c where c > 0 (a positive probability)
If each outcome k has a positive probability c, then the sum of the probabilities over all infinitely many outcomes would be:
Σ(k=1→∞)(ℙ(X=k)) = Σ(k=1→∞)(c) = c+c+c+...
This sum would diverge to infinity (c×∞ = ∞), which clearly contradicts the axiom that the total probability must sum to 1.
- Case 2: ℙ(X=k) = 0 (zero probability)
If each outcome k has a probability of 0, then the sum of the probabilities over all infinitely many outcomes would be:
Σ(k=1→∞)(ℙ(X=k)) = Σ(k=1→∞)(ℙ(X=k))(0) = 0 +0+0+... = 0
This also contradicts the axiom that the total probability must sum to 1.
3. The Axiom of Total Probability:
For any valid probability distribution, the sum (or integral for continuous distributions) of the probabilities of all possible outcomes in the sample space must be exactly 1. This axiom ensures that something in the sample space is certain to happen.
In summary:
- How: Because no single probability value (whether positive or zero) can be assigned to each outcome in an infinite discrete set such that the sum of all probabilities equals 1.
- Why: It directly violates the normalization axiom of probability, which states that the total probability of the entire sample space must be 1.
This impossibility holds for any countably infinite set (like integers, rational numbers). It's important to distinguish this from continuous uniform distributions over bounded intervals (e.g., a uniform distribution on [0, 1]), where the probability of any single point is indeed zero, but probabilities are assigned to intervals via integration, and the integral over the entire interval sums to 1. However, for a discrete set, there's no "interval" in the same sense to distribute the probability mass."
Post a Comment