Kinda controversial probability question?

Question

NO TWO SNOWFLAKES ARE ALIKE.

A container has an infinite number of snowflakes. I randomly pick 1 snowflake from this container. What is the probability that my snowflake is unique?

Answer 1

Fun question. I'm going to say 100%; your snowflake is unique, and this is my reasoning.

The set of snowflakes in the container is infinite. It is also countable, since picking out one snowflake at a time provides an algorithm that eventually reaches any given snowflake in the set. The set of natural numbers is also countable, so there exists a bijection between the set of snowflakes and the natural numbers. The natural numbers are unique under the equivalence relation "=", or strict equality. [To be overly pedantic, equality partitions the natural numbers into equivalence classes that consist of one element each and are isomorphic to the natural numbers.]

Now, randomly pick an element n from the set of natural numbers. By definition of equality, n is unique. By definition of bijection, the snowflake mapped to n is unique. Thus you have randomly chosen a snowflake from your container, and it is conclusively unique.

Answer 2

Well seeing that once you stick your 98.6 degree hands into the container, most of the snowflakes will melt, I'm not sure this is even possible.

But assuming your hands are cold enough, the probability the snowflake is unique is 1 since "No two snowflakes are alike". I'm not really seeing the controversy here.

If x[1] ≠ x[2] ≠ x[3] ≠ ... ≠ x[n] for all n
Then P(x[i] ≠ x[j]) = 1 for any i and j, i ≠ j

Answer 3

Well, the statement, "no two snowflakes are alike" is identical to, "every snowflake is unique." Since every snowflake in the bucket has the characteristic, "uniqueness," it must be 100% probability that the one you pick is unique.

Answer 4

Read this link to show that a reasearcher did find two identical snowflakes at the microscopic level. As the article continues, at the molecular level they probably weren't identical, but it's likely that nothing is identical at this level anyway.

http://www.abc.net.au/science/k2/moments/s1784760.htm

As for the actual probability question, I'd say 100%. For those people saying 0 because they're calculating 1/(a very large number), the probability is actually stated in the reverse so it should be "a very large number"/1.

Answer 5

It depends... if they are all fully formed snowflakes, then there is a 100% chance that your snowflake is unique. However, if the snowflake is not fully formed, there is a possibility that it may not be unique.

Answer 6

If your assumption is correct - that no two are alike - then the probability is 100% that your snowflake is unique. It doesn't matter how many are in the container.

Answer 7

Doesnt infinity stand for a VEEEEEEEEEEEEEEERY LAAAAAAAAAAAAAAAARGE real number? (If I remember my introduction to calculus) An actual number? And that the molecules of snowflakes in a fixed volume container are constant right? Mass, matter, energy cannot be destroyed or created. Therefore there is always a chance for the molecules to form together in the exact same way because the number of combinations stay constant. Though it will take a billion years(go through the entire container, find nothing, remelt snow, freeze again and go through again.) THERE WILL BE TWO SAME shaped snowflakes.

and... again... if I rememeber my intro to calculus.
... 1/inifinity = VEEEEEEEEEEEEEERY small number ... not zero

Answer 8

If you were to pick one snowflake out of infinite, then this expression can represent it:

1/∞, and by definition 1/∞ = 0

I would say 0, just because we are dealing with infinity.

Answer 9

I say 0.

Just because some person came up with the phrase "NO TWO SNOWFLAKES ARE ALIKE" doen't mean they've seen every snowflake and, therefore, know that it's true.

ps. I'm not trying to be negative. That's just my argument for the situation.