1
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0answers
82 views

Limit of constant function

I was reading a proof using Markov Chains in a finite state space $E$. Denote $p_{ij}(n) = P(X_n = j | X_0 = i)$. Since the state space is finite, then probability of landing somewhere in the state ...
6
votes
4answers
370 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
1
vote
1answer
62 views

In an infinity of choices, is it possible to guess the correct one?

So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in ...
1
vote
1answer
72 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
0
votes
1answer
26 views

Probability on the plane

Problem. On the Cartesian plane with origin O and x- y-axes, I randomly pick a point P. What is the probability that the line segment OP has a slope at least 1? Is the answer 1/4 or 1/2? answer = ...
2
votes
2answers
57 views

Is probability meaningful in cases of infinity?

Is it meaningful to speak of probability in cases of infinity? For instance, consider me having an infinite line of balls arranged in the manner: - Red, Green, Blue, Red, Green, Blue, Red....... ...
2
votes
2answers
259 views

Does The Monty Hall Problem Still Apply With Infinite Doors?

Here's been a bunch of questions on the Monty Hall problem, so I'll assume people know the basics. This answer helped clarify a few things for me, but talking with some colleagues yesterday, someone ...
6
votes
4answers
991 views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
1
vote
3answers
185 views

Creating the set of natural numbers

I am not a mathematician but an engineer, so I can read some basics of the language proofs are written in. Second I am bad in probability and infinity and my question covers both. So I like to ...
2
votes
1answer
146 views

What is the probability of guessing the right number $n$ from all numbers $\mathbb{N}$?

As we all know, $\mathbb{N}$ contains infinitely many numbers. What is the probability of guessing the right number $n \in \mathbb{N}$, i. e. what is $\frac{1}{\infty}$? It is clear that there is a ...
6
votes
2answers
2k views

Infinite expected value of a random variable

How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
8
votes
6answers
553 views

Is $\frac{1}{\infty}$ equal zero?

After reading this paragraph: A simpler version of this distinction might be more palatable: flip a coin infinitely many times. The probability that you flip heads every time is zero, but it ...
55
votes
6answers
4k views

Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?

In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only ...
4
votes
1answer
187 views

Maximizing gambling performance over the long run

Background. We can play a game in which we can put one dollar and get out $X$ dollars, where $X$ is 2 dollars with probability $p>1/2$, or zero dollars with probability $1-p$. We also assume that ...
0
votes
2answers
215 views

Resolving a paradox concerning an expected value

We have a coin that has a probability $p>1/2$ of coming up heads (and probability $1-p$ of coming up tails). We now play the following game: We start with a fortune of one dollar. We toss the ...
1
vote
0answers
50 views

Probability in infinitary logic

Let X be a random variable taking the value 0.2 with probability 0.2, 0.4 with probability 0.4 and 0.8 with probability 0.2 and 1.0 with probability 0.2. Using Infinitary logic I can ask the ...
0
votes
2answers
306 views

Proof of an infinite sum of probabilities [duplicate]

Possible Duplicate: Alternative Expected Value Proof If $X$ is a random variable that takes values in the range $\left \{ 1,2,3,4,5,6,\ldots \right \}$ how can I prove the following ...
5
votes
4answers
279 views

The limit of binomial distributed random variable

Edit (As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question) For infinite independent Bernoulli trials with probability ...
171
votes
14answers
21k views

Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? [closed]

Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who ...
6
votes
3answers
2k views

Probability and Infinity

If the probability of an event is $\frac{1}{\infty}$ and $\infty$ trials are conducted, how many times will the event occur — $0$, $1$, or $\infty$?