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.

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77 views

Probability of a disaster

Can you help me with this Markov chain problem? This should be simple, but my brain is not working well. Let $\{{x_t}\}_{t=0}^\infty$ follows a Markov chain. Each $x_t$ can take two values: {G,B} (...
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Getting exactly $n$ points from coin tossing [duplicate]

Possible Duplicate: A probability question Needing a little help with the following problem, A player tosses a fair coin and is to score one point for every head turned up and two points for ...
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356 views

probability density function of joint density functions

-UPDATED SOLUTION from sasha's hint- Thanks for checking my solution and see if i made any conceptual error. the question http://dl.dropbox.com/u/5681270/Screen%20Shot%202011-11-09%20at%201.40.47%...
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expected value of battery drawing problem

This is the question: In a box, George has $m$ batteries of which $n$ are dead. He tests them randomly and one by one. Every time that a good battery is drawn, he will return it to the box; every ...
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Expected time of last bus left

I came across the following problem in a probability exercise, and got confused. The problem is the following. Problem: In a bus station, assume the expected time of next bus arriving is a ...
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Find standard deviation

The assignment is given: In a heat regulated room, we have two temperature limits $T_{\text{min}} = 16$ and $T_{\text{max}} = 24$. If the temperature is between $T_{\text{min}}$ and $T_{\text{max}}$...
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Prove (if true): $P(A|C) = \sum_{i} P(A|B_i)P(B_i|C)$

I think the statement $P(A|C) = \sum_{i} P(A|B_i)P(B_i|C)$ is true (where the events $B_i$ form a partition of the whole space). Is it, and why? Thanks!
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Need a pointer on conditional probability of a fair coin toss

A fair coin is thrown $n$ times. Show that the conditional probability of a head on any specified trial, given a total of $k$ heads over the $n$ trials, is $\frac{k}{n}$ ($k > 0$). I immediately ...
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1answer
163 views

Restricted random walk in $\mathbb Z^3$

What is the proability to return to the origin, for a uniform random walk on the integer lattice in $\mathbb Z^3$, if we are restricted to $x \geq 0$? I.e. if we try to step into a negative x-...
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Top 3 of 4 Dice Rolls

I'm trying to prove why the mean of the distribution of sums of the top 3 out of 4 fair 6 sided dice is rolls 12.25. Anybody who's rolled a D&D character knows the idea. $r_n = Rand([1,6])$ $x =...
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Find the cdf of X

A point is chosen uniformly at random from the circumference of a circle of diameter 1. Let X be the length of the chord joining the random point to an arbitrary fixed point on the circumference. Find ...
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poisson-binomial mixture tail bound

Let $X \sim \operatorname{Binom}[(n,p)]$ and $Y \sim \operatorname{Poisson}[f(X)]$, where f is a convex function. Are there any good tail bounds for $Y$? For instance, are there any Chernoff-style ...
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Conditional probability for the sum of drawn cards. Where am I going wrong?

I have the following practice problem. From a deck of five cards numbered 2, 4, 6, 8, and 10, a card is drawn at random and replaced. This is done three times. What is the probability that the card ...
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Relative entropy between singular measures

Usually, to define relative entropy between two probability measures, one assumes absolute continuity. Is it possible to extend the usual definition in the non absolutely continuous case?
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How to express disjunction in Probability Theory?

Joint probability is usually expressed with a comma: $P(A=a, B=b) $ meaning: The probability of random variable A having value a and random variable B having value b But what would be the notation of ...
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Calculating the probability that a cord makes a knot

I use earphones to listen to music. The cord connected to the earphones often gets entangled in my pocket and makes a knot, which I always find hard to untangle. How can I define and calculate the ...
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1answer
513 views

Bernoulli trials hypergeometric relation

The problem is: In a sequence of independent bernoulli trials let X be the number of successes in the first m trials, and Y be the number of successes in the first n trials, $m<n$. Show that the ...
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1answer
323 views

A Problem Related to Monkey Typing

I got a problem which is (I think) similar to the monkey typing problem: Suppose we flip a coin $n$ times to obtain a sequence of flips $X_1, X_2, \dots, X_n$. A streak of flips is a consecutive sub-...
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probability of number of draws

A box contains 8 tickets. Two are marked 1, two marked 2, two marked 3, and two marked 4. Tickets are drawn at random from the box without replacement until a number appears that has appeared before. ...
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1answer
682 views

Counterexample to Jensen's inequality

This appeared in an exam I took. The question asked us to give an example of a convex function $g: \mathbb{R} \longmapsto \mathbb{R}$ and a measure $\mu$ on $\left(\mathbb{R}, \mathscr{B}(\mathbb{R})\...
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Absolute continuity of a distribution function

This appeared on an exam I took. $Z \sim \text{Uniform}[0, 2\pi]$, and $X = \cos Z$ and $Y = \sin Z$. Let $F_{XY}$ denote the joint distribution function of $X$ and $Y$. Calculate $\mathbb{P}\left[...
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chances of getting three of one kind and four of another out of seven dice

There are several questions similar to this one but after reading those, I am still very confused. I also did a similar problem of this one and I think I got it, but then I got stuck again. So if ...
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The probability density function of the ratio of two normal R.V.s

I'm looking for some help with this probability problem. Here's the question: Suppose that $X$ and $Y$ are independent standard normal random variables. Show that the probability density function ...
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Maximizing a function containing an integral

Problem. Let $\rho\colon[-1,\infty)\to\mathbb{R}$ be a function such that $$\int_{-1}^\infty\rho(x)\,dx=1.$$ Let $G\colon[0,1]\to\mathbb{R}$ be a function that is defined with $$G(f) := \int_{-1}^\...
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Expressing “Probability that #successes is an even number” mathematically

Needing a little help with my probability concept. Here's the question: An urn contains $10$ red balls, $20$ green balls and $30$ blue balls. Each trial consists of drawing a ball from the urn with ...
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351 views

Confusion about joint distribution between two independent random variables

Let $X,Y$ be independent random variables, uniform on $(0,1)$. a) $P(X+Y>1.5)$. b) $P(X>Y \mid X>1/2)$. c) $P(\tan^{-1}(Y/X)<t)$ for all $0<t<\pi/2$. e) $E(\tan^{-1}(Y/X))$. I ...
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1answer
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Samples in the convex body vs. samples on the convex surface

Let $K$ be a bounded convex body in $\mathbb{R}^n$. Suppose we have a sampler $\mathcal{S}_1$ that can generate points uniformly distributed in $\mathrm{int}K$, and another sampler $\mathcal{S}_2$ ...
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About joint probability divided by the product of the probabilities

Let $X$ and $Y$ be two events. So $P(X)$ is the probability of $X$ happens, and $P(Y)$ is the probability of $Y$ happens. So $P(X,Y)$ is probability of both $X$ and $Y$ happen. So what is the ...
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1answer
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One container contains 6 red and 4 white balls, while a second container contains 7 red and 3 white balls

One container contains 6 red and 4 white balls, while a second container contains 7 red and 3 white balls. A ball is chosen at random from the first container and placed in the second. Then a ball is ...
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Central Limit Theorem and sum of squared random variables

This is a two-part question. Suppose I am drawing random variables $X_i\sim A$, $1\leq i \leq n$ where $A$ is a zero-mean, finite variance $\sigma_A^2$, symmetric probability distribution having ...
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1answer
188 views

Probability of uniform distribution

If I have an array of $n$ numbers in the range [0,1) and have the following events for $y$: for $i=1$ to $n$ : generate a new number $y$ in [0,1) uniformly and independent of previous if $y\ge0.5$ ...
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A weather station recorded the temperature every 10 minutes

I need your help to understand this homework. A weather station recorded every 10 minutes the temperature. X is the minimum and Y the maximum temperature in a day. between the 1st and 30th April the ...
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4answers
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Probability of defeating enemy (info on distributions added)

I have 27 hit points and my opponent has 50, and the winner is the player that reduces the other player's hit points to 0 first. My expected damage inflicted per round is 5. My expected damage taken ...
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Probability question: optimal strategy

I am really confused about how to think about this question. It was presented as a challenge by a peer. Two people seek to kill a duck at a location $Y$ meters from their origin. They walk from $x=0$...
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Calculating mu and sigma (μ and σ) of a normal random variable

Let X be a normally distributed variable with unknown parameters μ and σ (sigma). If we know that P (X ≥ 75) = 0.7291 and P (X ≥ 83) = 0.7764. With the information given Is it possible to determine ...
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1answer
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Conditional probability of baseball series/ combinations of baseball series

1 The first two and the last two games of the series are scheduled to be held in St. Louis while the middle three are scheduled to be held in Texas. How many possible outcomes are there in which St. ...
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Working out the variance of the Poisson distribution

My lecturer in his notes uses this definition of Poisson distribution: $$ P_{X}(t)= \exp( \lambda (t-1)) $$ You differentiate once and set equal $t=1$ to get $E[X]=\lambda$, but in the notes to get ...
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1answer
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Probability about a geometric distribution

Bill, Mary and Tom have coins with respective probabilities $p_1,p_2,p_3$ of turning up heads. They toss their coins independently at the same times. What is the probability that neither Bill nor Tom ...
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872 views

Weak*-topology and probability measures

Let $Y$ denote a separable metric space, and $\mathcal{P}(Y)$ the set of probability (understood as countably additive) measures on the Borel $\sigma$-algebra of $Y$. The weak*-topology says that a ...
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1answer
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Estimate a upper bound of IQ scores

Suppose the IQ scores of a million individuals have a mean of 100 and an SD of 10. a)Without making any further assumptions about the distribution of the cores, find an upper bound on the number of ...
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Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$

I don't really know how to start proving this question. Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite. Show that $E(\xi\mid\xi+\eta)=E(\eta\mid\...
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424 views

Represent a value of conditional expectation by unconditional expectation?

In my previous question, I asked why $$ E(Y |X = x) = \int_\Omega Y (\omega)P(d\omega|X = x) = \frac{\int_{X=x} Y (\omega)P(d\omega)}{P(X = x)} = \frac{E(Y \, 1_{(X=x)})}{P(X = x)} $$ when $X$ is ...
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With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?

I've got a balls and bins problem. Suppose I throw $n$ balls uniformly at random into $n$ bins. What is the probability that exactly $k$ bins end up with exactly $1$ ball? I know this seems a ...
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2answers
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Expectation of Random Walk

At each time step, I have 1/2 probability of walking one step to the right, and the same probability of walking one step to the left. Let X be the random variable corresponding to the final ...
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1answer
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Coin-tossing game

I'm doing a really old IB Math Higher Level exam question about an interesting coin flipping game. I have solved some of the question (I think?), but I'm stuck now. Here it goes: An unbiased coin ...
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1answer
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Probability of Monkey typing keyboard

A monkey types at a 26-letter keyboard with one key corresponding to each of the lower-case English letters. Each keystroke is chosen independently and uniformly at random from the 26 possibilities. ...
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Expected number of runs in a sequence of coin flips

A coin with heads probability $p$ is flipped $n$ times. A "run" is a maximal sequence of consecutive flips that are all the same. For example, the sequence HTHHHTTH with $n=8$ has five runs, namely H, ...
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208 views

Optimal number of answers for a test with wrong-answer penalty

Suppose you have to take a test with ten questions, each with four different options (no multiple answers), and a wrong-answer penalty of half a correct answer. Blank questions do not score neither ...
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Probability with a collection problem

A collection of tickets comes in four colors: red, blue, white, and green. There are twice as many reds as blues, equal numbers of blues and whites, and three times as many green as whites. I choose 5 ...
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Probability of G Good elements and B Bad elements.

A box contains 3 red balls, 4 blue balls, and 6 green balls. Balls are drawn one-by-one without replacement until all the red balls are drawn. Let $D$ be the number of draws made. Calculate: a)$P(D\...