From an infinite set of numbers, is it possible to select the same number twice at random?

Question

Before you jump to the YES conclusion, think that I'm not talking huge amounts of numbers; I'm talking INFINITE numbers. The chance of choosing the same number twice is dictated in part by the number in the set, right? So infinite set, what can we say?

Answer 1

Simple answer:
A closed set of numbers, say [0,100] and you have two selection chances. Using probability you would a 1 in 50 chance of picking the same number because:
Probability (P) = Chances (C) / Total (T) or
P = C/T
therefore:
P = 2/100
P = 50

But as x goes to infinity, the denominator would become large so you would be in a limit situation so:
P = Limit as T approaches infinity of C/T
would become zero or a vertical asymptote at zero.

Complex answer:
Study topology and set theory.

Answer 2

Technically yes. But the chances are so close to zero that you might as well say no.

It HAS to be yes. When you are doing your second selection the chances of choosing the same number is the same as any other number. If the chances were 0 of picking the same number again then the chances of picking any other number would be 0. This is not possible because it says that you cannot pick a number at all. So even though it may look like 0, in fact there is a chance.

Using Carmicheal99's equation from below, P = C/T.

As T tends to Infinity the probability becomes so small that it appears to be 0. But it never actually BECOMES 0. 0 is the limit.

Answer 3

Let n = number of numbers in the set. After choosing any partifcular number in this set, the odds of not choosing it again with any random choice is:

(n-1) / n, or

1 - 1/n

Let m be the number of choices made. Then the odds of not having chosen the orginal number after m tries is:

(1 - 1/n)^m

so that the odds of chosing the original number is:

1 - (1-1/n)^m

The limiting case is where n & m -> infinity. Therefore, the answer is:

1 - 1/e, or approximately 63%. (e is Euler's number)

The odds of picking the original number again in an infinite set of numbers after an infinite number of tries is about 63%. On the other hand, the odds of picking the same number twice at any time from an infinite set of numbers after an infinite number of tries is 100%. To illustrate this last point, let's say that I have a set of 10 different numbers. I make 10 random choices. The only way I can avoid picking a number twice is to pick each one exactly once after 10 tries. The odds of that drops to 0 with an infinite set of numbers.

Answer 4

It depends on the probability distribution.

If it is the uniform distribution on the interval [0,1], then the answer is no, since the probability of selecting any particular number is 0.

On the other hand, consider the number of flips for a coin to land on heads. You might see a million, a billion, any number of tails before you see a heads, so the set of possible outcomes is infinite. Another way to describe this distribution is P(X = n) = 1/2^n for n > 0. This is a probability distribution because 1/2 + 1/4 + 1/8 + 1/16 + ... = 1. Because individual outcomes have a positive probability, it is possible for them to recur.

Answer 5

There's a subtlety here that's sometimes known as the St Petersburg paradox.

In order for there to be a finite expectation, the series must converge. It does not for the process that you've described.

Consider also, that for a continuous distribution the probability of getting the same value twice is zero because there's zero area under a single point.

Keep digging into this and you'll get to Zeno's paradox and some of the fundamental aspects of limits and continuity.

Consider the following: approach people and ask them to give you any integer. Keep doing this and you will most likely get a repeat fairly soon because people's choices are hardly random.

Answer 6

Imagine a perfect random number generator.
In a perfect random number generator, if you select two numbers, the second number is totally independent on the first.

Imagine setting the random number generator so that it never selects the same number twice in a row.
Then select two numbers.
The first number is truly random.
However, the second number is not random because its selection pool depends on the first number chosen.

Therefore, it is possible to select the same number twice a random from an infinite set of numbers. In fact, if you select from that infinite set of numbers enough times (also infinite), you have a near certainly of selecting the same number twice. p --> 1 as n --> infinity.

Answer 7

I am not sure there is such a thing as infinite.
There are things approaching infinite such as stars in the universe or water molecules in the ocean. If we use the word uncountable instead then the answer is yes. Actually considering the
methods of selecting numbers "randomly" currently available the answer is even more yes but still highly unlikely.

Answer 8

Yes it is always possible to select two same numbers at random from infinite set, if you are talking selecting number manually then the probability might be zero, but if u r talking about doing it programatically then yes it is indeed possible.

Answer 9

Yes, it's possible. Someone said that the probability of it happening is 0, but that's not the same thing is it being impossible.

Counterintuitive, but true: the probability that tomorrow's high will be exactly 32 degrees (and not 32.000001 or 31.999) is also 0, but it's possible for it to happen. If it's impossible to be exactly 32 degrees, it's also impossible to be 32.00000000001 degrees, and every other temperature; this would imply that every temperature is impossible to occur, which is clearly false.

Answer 10

Yes.