Articles of probability

Probability: What Is The Average Number of Steps?

This question already has an answer here: Expected number of steps till a random walk hits a or -b. [duplicate] 1 answer

“off by 1” lottery probability

Suppose there is a lottery such that 6 random balls are chosen from a set of 50. The balls are numbered 1 thru 50. The lottery officials determine that lottery ticket sales are sluggish so they want to make it easier to win so they decide to allow “off by 1” for each of the […]

how do i calculate the threshold value of the following problem?

Sorry for asking this but this math problem has got me confused. How do i go about calculating the threshold value of this problem? Consider that I have an asset worth $2000. There are two independent threats. The first occurs with probability 0.05 and would reduce the value of the asset to $100, while the […]

Is a binomial a sum of Bernoulli random variables?

I know that, if $X_1,\ldots,X_n\sim \text{Ber}(p)$ are independent, then $X_1+\ldots+X_n\sim\text{Bin}(n,p)$. My question is: if $X\sim \text{Bin}(n,p)$, is it true that there exist independent random variables $X_1,\ldots,X_n\sim\text{Ber}(p)$ such that $X(\omega)=X_1(\omega)+\ldots+X_n(\omega)$ for all $\omega\,$? EDIT: I would like an analytic proof of this fact (if it is true).

Independence of sigma algebra

I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof. If $X_1, X_2$ and $X_3$ are pairwise independent random variables, then if $Y=X_2+X_3$, is $X_1$ independent to $Y$? (One can think of an example where the $X_i$ […]

Ratio between $k$th highest number among $n$ and $n+1$ samples

Let $n\geq k$ be fixed positive integers, and let $X$ be a distribution on $[0,1]$ that is not the constant $0$ distribution. Let $E_n$ denote the expected value of the $k$th highest value among $n$ independent samples from $X$, and $E_{n+1}$ the expected value of the $k$th highest value among $n+1$ independent samples from $X$. […]

Rooks Attacking Every Square on a Chess Board

8 rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column. Compute the probability that every square is occupied or attacked by at least 1 rook. The first step I took was to state that there are $64C8$ ways […]

Probability of predicting, then throwing, a particular multiset for 5 dice.

My friend shared with me a story that after losing to his SO at Yahtzee, before they put the game away he just randomly predicted he would roll four 5’s and a 1. He then got that roll and freaked out. I wanted to calculate exactly how likely that was but my combinatorics knowledge isn’t […]

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability densities: $$g_0(y)=f(\delta(y),\mu_0,\lambda_0)f_0(y)$$ and $$g_1(y)=f(1-\delta(y),\mu_1,\lambda_1)f_1(y)$$ where $f_{0,1}$ and $g_{0,1}$ are some continuous probability denstiy functions on $\mathbb{R,}$ $\mu_{0,1}$ and $\lambda_{0,1}$ are some […]

Probability of particular subset of balls occurring in a larger set chosen from a total?

If I have 3 balls chosen out of 6 (2 blue, 2 red, and 2 green) with replacement, what is the chance a chosen ball will be blue? How about two blue balls? I think I can reason this out, but I’m curious about the approach using permutations and combinations. So far, I’m thinking I […]