Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

5
votes
2answers
574 views

New Two Children problem

The well known "Two Children problem" is answered in " In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?" What about this variant: ...
2
votes
4answers
353 views

Two weighted coins, determining which has a higher probability of landing heads

A friend of mine asked me the following question, and I am not sure how to solve it: You are given two weighted coins, $C_1$ and $C_2$. Coin $C_1$ has probability $p_1$ of landing heads and $C_2$ ...
5
votes
2answers
429 views

Coin Betting Expectation

Suppose I have a biased coin with probability of heads p, and tails q=(1-p). It is then used in a game which lasts at most N tosses, and start with a stake of £1. Each time the coin is tails my money ...
4
votes
1answer
188 views

Discovering properties of a graph by means of random walk

Suppose I have a regular, undirected, non-bipartite, finite, connected graph on $N$ vertices. Some fraction $\frac{c}{N}$ of the vertices are coloured gold, the rest are coloured black. If I let you ...
7
votes
1answer
1k views

Do the moments characterize a distribution with compact support?

Given a random variable $X=(X_1,...,X_p)$ with $P(X \in M) = 1$ for compact $M$, do the values of $E[X_1]$, $E[X_2], ..., E[X_1^2], E[X_1 X_2],..., E[X_1^3],...,E[X_1 X_2 X_3]... $ determine the ...
18
votes
2answers
2k views

3 other extensions of the Secretary Problem

In the classical secretary problem (also known as Marriage, Sultan's Dowry, Gogol problems), 1) There are $n$ candidates ordered from the best to the worst (no ties). We know $n$. 2) The candidates ...
5
votes
2answers
2k views

Matching Hats AND Coats Problem

Suppose $N$ men throw their hats in a room AND their coats in an other room. Each man then randomly picks a hat and a coat. What is the probability that: None of the men select his own hat and his ...
1
vote
1answer
1k views

On the tightness of Chernoff bounds for sum of Poisson trials

For the sum $X$ of independent 0-1 random variables $X_i$ ($0 \le i \le n-1$) with $Pr(X_i)=p_i$, namely $X=\sum_{i=0}^{n-1}{X_i}$ the following Chernoff bound holds, $$ Pr(X \ge (1+\delta)\mu) \le \...
1
vote
1answer
2k views

Differentiation of expectation of a nonlinear function

I encountered the following problem and need some help. Let $X$ be a continuous random variable. (You can assume $X$ to be very nice: it has a smooth density function with bounded support, bounded ...
4
votes
1answer
1k views

How can I get better at this certain kind of probability problem?

I'm studying for an onsite interview with Google for a product manager position. While looking at interview questions online, I've realized that I really need to brush up on the probability and ...
2
votes
2answers
310 views

Intuitive explanation of why we integrate to find expectation

If I have a random variable X with probability mass function, $f(u)$, would somebody give me an intuitive reason as to why the expected value of X is $\int^t_0 x f(x) dx$ ?
2
votes
2answers
276 views

Describing Bayesian Probability

I'm a CS major doing some work with image recognition in which I use Bayesian probability. I have to give a presentation on my work, and while I have no problem describing the CS portion, I'm less ...
4
votes
1answer
497 views

Probability of getting a royal flush with four wild-cards

I am trying to calculate the probability of getting royal flush, if four 5's are wild cards that can be of any suit. I get that the probability of the first card I am picking is $\frac{24}{52}$, but ...
1
vote
3answers
245 views

How many times do I need to toss a coin

I have a homework question that's driving me crazy: How many times do we need to toss a coin so that with prob. .9 the discrepancy of the relative frequency of appearance of heads from 1/2 is less ...
3
votes
1answer
245 views

Sum of two independent random variables

If A and B have the same distribution, e.g. Poisson, and they are independent. Is $P(A + B = k) = P(A = k)P(B = k)$ ? If so, why? If not, what is the correct way to calculate $P(A + B = k)$?
10
votes
3answers
2k views

probability distribution of coverage of a set after $X$ independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
3
votes
3answers
566 views

Finding probability of an unfair coin

An unfair coin is tossed giving heads with probability $p$ and tails with probability $1-p$. How many tosses do we have to perform if we want to find $p$ with a desired accuracy? There is an obvious ...
2
votes
2answers
215 views

A lower bound on the probability that a variable is 3/2 times the expected value

Surely many of these are coming now. I'm reviewing for final exams, and came across this problem. I have a list of length $n$, and some process that reduces the length of the list by expected size $\...
4
votes
1answer
318 views

A question about linear regression

Select $n$ numbers from a set $\{1,2,...,U\}$, $y_i$ is the $i$th number selected, and $x_i$ is the rank of $y_i$ in the $n$ numbers. The rank is the order of the a number after the $n$ numbers are ...
0
votes
3answers
400 views

probability density functions of random variables x and y

the question asks for the density of the smaller of $X$ and $Y^3$, $X$ and $Y$ both being exponentially distributed independent random variables with densities $ae^{-ax}$. I think I know that I have ...
0
votes
2answers
2k views

finding a probability of a deuce in a tennis game

I have a probability of Player1 (1) winning a point with a probability of $\frac{3}{4}$, P2 (2) then then has a probability of winning of $\frac{1}{4}$. I need to find the probability that a deuce ...
2
votes
1answer
658 views

Reviewing for exam: Chernoff bounds for a series of random variables

I have a series of random variables, where the expected value is $\frac{n}{4}$. I want to prove, with Chernoff bounds, that the probability that the actual value is less than $(1 - \epsilon)\frac{n}{...
4
votes
1answer
3k views

Simple probability question, balls and bins

This is a simple question I came across in reviewing. I am wondering if I got the correct answer. The question is simple. You have $n$ balls and $m$ bins. Each ball has an equal probability of ...
4
votes
1answer
478 views

Bertrand's paradox (statistics)

I just learned about Bertrand paradox in today's class, and am very shocked. I was wondering if there are indeed only 3 (known?) unique ways of picking a chord in a circle at random, with 3 different ...
1
vote
4answers
465 views

Can't determine whether problem involves the binomial distribution or not

The question is basic: If it's sunny an average of 5 days a week, what's the probability that it'll be rainy 2 days out of 3? Assume unrealistically that the weather on a given day is independent of ...
0
votes
1answer
547 views

Chernoff bounds on a random variable

I have some random variable with an expected value of $n/8$. I want to use Chernoff bounds to show that with some high probability, the actual value is at least $(1 - \epsilon)n/24$. How could I go ...
1
vote
0answers
855 views

Use of covariance matrix for the confidence interval

I have a number of explanatory variables $x_1,...,x_n$ and an outcome variable $y = f(x_1,...,x_n)$. Here $f$ is assumed to be known (estimated). I heard that for a confidence interval for $y$ one can ...
0
votes
1answer
97 views

Insert into a set based on a sequence of coin flips. Conditional probability?

I am studying for an exam and found the following problem. Let $L = l_0, l_1, ... , l_{n+1}$ be a list of items. For each item from 0 to $n+1$, we flip a coin (fairly). We add item $l_i$ (...
1
vote
2answers
2k views

pdf of combination of two distinct exponential random variables

Say X and Y are two independent random variables with exponential density\begin{split} f_{X}(x) = a e^{-ax}\end{split} and \begin{split} f_{Y}(y) = b e^{-by}\end{split}, then what is the probability ...
21
votes
3answers
649 views

Probability distribution for the remainder of a fixed integer

In the "Notes" section of Modern Computer Algebra by Joachim Von Zur Gathen, there is a quick throwaway remark that says: Dirichlet also proves the fact, surprising at first sight, that for fixed $...
6
votes
1answer
1k views

How many steps does it take the computer to solve a Sudoku puzzle?

We all know what Sudoku is. Given a Sudoku puzzle, one can use a simple recursive procedure to solve it using a computer. Before describing the algorithm, we make some definitions. A partial solution ...
1
vote
1answer
117 views

probability - what is $|X|$?

In probability , Let $X$ be an independent random variable $X$. When someone writes $|X|$-what does he mean? Thank you.
3
votes
2answers
240 views

Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't?

Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't?
1
vote
2answers
406 views

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?
107
votes
11answers
165k views

What is the best book to learn probability?

Question is quite straight... I'm not very good in this subject but need to understand at a good level.
6
votes
2answers
1k views

Collisions in a sample of uniform distribution

Asked at a Microsoft interview: Assume you have a uniform distribution (can be discrete or continuous) of size X and you randomly select a sample of size Y. 1) What is the probability in terms of X ...
3
votes
4answers
959 views

Probability of a Chord Passing Through Two Concentric Circles

One of my friends gave me a problem that stumped me... You have two concentric circles, one with a radius of 1 and one with a radius of 2. What is the probability that a random chord will pass ...
5
votes
2answers
643 views

Central Limit Theorem

If N is a poisson random variable, why is the following true? It is from "Probability and Stochastic Processes" by Yates, page 301, equation 8.2 to 8.3 $$ P\left(\left|\frac{N-E(N)}{\sigma_N}\right| ...
1
vote
1answer
283 views

Probability of having at least 'k' marbles specific to each of 'm' bags filled by sampling with replacement

I'm going to rewrite my original question to make it a bit clearer: Assume I have some set $P$ with $||P|| = N$ unique elements. I also have $S$ multisets, $(m_1, ..., m_S)$, of cardinality $L$, ...
1
vote
2answers
2k views

Prove that P(A|B,C)=P(A|C,B) after applying the Bayesian updating process

Prove that P(A|B,C)=P(A|C,B) after applying the Bayesian updating process. That is, prove that the order in which the information is presented does not matter.
3
votes
1answer
2k views

How do you compute the steady state probabilities of a continuous time markov chain?

Given a markov model with transition rate matrix $Q$, where the probability of each state at time $t$ is given by $P(t) = P(0)e^{Qt}$, where e is the matrix exponential, the steady state probabilities ...
1
vote
2answers
101 views

probability -one more question about expectation

Yesterday I asked you whether there's a situation where $E[X]$, $E[Y]$ defined, but $E[X\cdot Y]$ ($E[Z]$) is $\infty$ or even Diverging? You claimed that the answer is NO and it is not ...
3
votes
2answers
207 views

probability -Diverging expectation

As I keep reading probability books, there are always some issues that no one considers. For example, for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega )=X(\...
2
votes
2answers
561 views

Collisions in random walk in $\mathbb{Z}^n$

Given a set $S$ of $r$ points in $\mathbb{Z}^n$, $S=(p_1,p_2,p_3.., p_r)$ , each a starting point for random walk with step size 1. What is the probability they will all eventually meet at the same ...
3
votes
1answer
1k views

Are there any dependent and non-correlated random variables so that $E[X]\neq0$ and $E[Y]\neq0$?

there's a something that I though about that Disturbing me for quite some time now: Let $X$ , $Y $ be a random variables. I call random variables independent if for every possible value of $x_{i}$ ...
5
votes
2answers
2k views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
7
votes
2answers
561 views

How is a Halton sequence related to a Latin hypercube?

I currently use a Halton sequence to choose parameter sets for a prognostic model (e.g. using metabolic rate and protein content parameters to predict growth rate). From my understanding, both a ...
1
vote
2answers
5k views

What is the probability of success given two trials with a 1/16 chance each?

I have never been able to wrap my head around probability, and I often find that my intuition is wrong. In this case, I don't even have intuition as to where to begin. If I have two trials, each ...
0
votes
1answer
314 views

Probability of intersection of two geometrical figures in bounded space?

I'd like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures $F_1$ and $F_2$ of area $A_1$ and $A_2$, respectively, that are have a random ...
2
votes
1answer
184 views

Inequality involving random variables

I have random variables $X_1,\ldots,X_n \in \{0,1\}$ which are dependent and $$\mathbb{P}[X_i=1 | X_1=x_1, \ldots, X_{i-1}=x_{i-1}, X_{i+1}=x_{i+1}, \ldots X_{n}=x_n] \geq p$$ for any $x_1,\ldots,x_{i-...