Articles of probability

Is this a Delta Function? (and Delta as limit of Gaussian?)

I have a set of users that generate calls. If I assign the same probability to each user, they have identical call generation probability which can be defined as $\delta$. These callers are chosen uniformly among the set of users. At the end of the generation process, the representation of the probability density function of […]

Probability of drawing cards in ascending order

Given 200 cards where each card has a unique number from 1 to 200. We randomly pick 30 cards (the order we pick them matters). What is the probability the unique numbers of the cards we pick are in ascending order?

Find the expectation

A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ? The answer is $\large \frac{ca}{a+b}$. How to approach this one?

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number value while the person who has a larger number will halve their number value. Disregard any draws. For example, if […]

Expected number of runs

Let $S[16]$ be a binary array i.e, elements of $S$ are 0/1 with elements $S[i]$ are taken uniformly and independently form $\{0,1\}$. Let $k$ be a random element taken uniformly from $\{0,1\}$. I have run the following process. Take $a,b,c$ randomly and independently from $[1,16]$. Change $k=k+S[a]$ $S[b]=k+S[c]$. Run the whole process (1,2,3) until all […]

Prove these two conditional probabilities are equivalent

I saw people using such equivalence $$P(X|\mu) P(\mu | D) = P(X,\mu|D)$$ how to prove it is valid? My attempt: \begin{align} P(X|\mu) P(\mu | D) &= P(X|\mu) \frac{P(\mu,D)}{P(D)}\\ &= P(X|\mu) \frac{P(D|\mu) P(\mu)}{P(D)}\\ &=\frac{P(X,\mu) P(D|\mu)}{P(D)} \end{align} where $P(X,\mu|D) = \frac{P(X,\mu,D)}{P(D)}$. I have stuck here, couldn’t figure out why $P(X,\mu,D) = P(X,\mu) P(D|\mu)$. Edited Additional: If instead […]

Probability to find the sequence “Rar!” in a random (uniform) bytes stream at a position $\le n$

I have random (uniform distribution) stream of bytes (integers from $0$ upto $255$). What probability $p(n)$ to find the sequence “Rar!” (ASCII) at a position $\le n$ from the start of the stream?

Conditional Expectation of a Poisson Random Variable

Hello I’m having difficulties with the following problem: Let X be a Poisson random variable with parameter $ \lambda $. Find the conditional mean of X given X is odd. What I tried this: A = [X is odd.] Back to the basic idea E[X|A] = $\displaystyle\sum\limits_{x=0}^\infty xP_{X|A}(X=x|A=a) = \displaystyle\sum\limits_{x=0}^\infty x \frac{P_{X,A}(X=x,A=a)}{P_{A}(A=a)} $ I actually […]

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them anymore). The author purports to bound the probability of the Goldbach Conjecture being false as $\approx 10^{–150,000,000,000}$ What is mathematics to make of […]

When to stop in this coin toss game?

Suppose you play the following game: You toss a fair coin. If you get heads, a hundred dollars are added to your reward. If you get tails, however, the game is stopped and you do not get anything at all. After each throw you can decide, whether you want to take the money or keep […]