This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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1answer
25 views

Same birthday question: 2 different variants have same answer?

How many people must there be before the chances that someone has the same birthday as you do is at least 0.5? How many people must there be before the chances that at least two people have a birthday ...
0
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3answers
67 views

How do you deal with fractions in a binomial?

If I have something like this $$\binom{\frac{x}{k}}{\frac{y}{k}}$$ (where there are two fractions in a binomial but they have the same denominator) can I simplify this at all?
3
votes
1answer
66 views

What's the probability of winning a raffle with extra lives?

My local ACM club has prizes at every meeting. You get 1 life for showing up at the meeting, and 1 extra for solving the programming challenge. The script chooses one of the remaining people, ...
0
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1answer
161 views

Help with Durrett 5.7.8

I've been working on Durrett Exercise 5.7.8 without any avail: Let $S_n$ be the total assets of an insurance company at the end of year $n$. In year $n$, premiums totaling $c>0$ are received and ...
0
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1answer
38 views

How do I figure out what the distribution of X is?

I'm trying to figure out what distribution X follows in this question: Of the people passing by a luxury clothing shop 0.4% make a purchase. Let X be the number of those who will make a purchase ...
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2answers
75 views

Probability clarification $\chi^2$ distribution table

I'm having some trouble understanding the solution of this probability question. Ammeters produced by a manufacturer are marketed under the specification that the standard deviation of gauge ...
0
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1answer
44 views

What is the probability that an arrangement of 'FACETIOUS' begins with A and ends with I?

In how many ways can the letters of the word FACETIOUS be arranged in a line? What is the probability that an arrangement begins with A and ends with I ? I understand the first part which will be $9!=...
0
votes
1answer
45 views

finding p value of a test , no population mean given.

How do i find the p- value of this sample ? 10, -1,6,7,-5,-5,2,-3,8,9,-2 I do not know the population mean , only that it came from a normal distribution. My problem finding Z value first then i ...
0
votes
1answer
22 views

Disjoint Union and Total Probability Theorem question

There is some clarification I need with respect to a question on the following formula. For any sequence of events $B_1,B_2,\dotsc,B_n$, $$\mathbb P\left(\bigcup_{i=1}^nB_i\right) =\sum_{i=1}...
0
votes
0answers
27 views

conglomerability

As I understand it, conglomerability tells me that, if $\mathcal{S}$ is a (potentially uncountable) partition, then $$ \Pr(A) \in ConvexHull[\{ \Pr(A \mid S_i) : S_i \in \mathcal{S}\}] $$ And, ...
0
votes
0answers
30 views

Find the covariances of a multinomial distribution

If $(X_1,\cdots, X_n)$ is a vector with multinomial distribution, proof that $\text{Cov}(X_i,X_j)=-rp_ip_j$, $i\neq j$ where $r$ is the number of trials of the experiment, $p_i$ is the probability of ...
2
votes
1answer
67 views

Is $X^2$ independent from $XY$ where $X$ and $Y$ are standard normals?

I'm thinking they can somehow be expressed as functions of $X-Y$ and $X+Y$, but I haven't quite found out how. Bonus questions: Is it correct that they are both Chi square distributed? And so, ...
0
votes
3answers
63 views

What is the probability that he picks up a blue pen?

A boy has color weakness, so he is not good at distinguishing blue and red. There are $60$ blue pens and $40$ red pens randomly in a box. Given that he picks up a blue pen, there is a $60$% chance ...
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0answers
11 views

Reparametrization in regards to Fisher information [duplicate]

I'm trying to understand equation 5.11 from Lehmann and Casella's Theory of Point Estimation (2nd edition) which is presented without proof. It states that if $I(\theta)$ represents the Fisher ...
0
votes
1answer
35 views

Urn balls multinomial with joint probability function

An urn contains $w$ white, $b$ black and $r$ red balls. $n$ extractions with replacement are made. $X_w$, $X_b$ and $X_r$ are the random variables representing the number of white, black and red balls ...
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votes
2answers
44 views

Poisson Coin tossing on to a page question

I am having difficulties with a question that goes as follows: "The mean no. of misprints on a page of area 400 $cm^2$ is 10, A silver dollar of area 20 $cm^2$ is tossed on the page at random. Find ...
1
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1answer
27 views

$L_p$ norm $\leq L_2$ norm for $1\leq p\leq2$ for Random Variables

Let {$X_i;i\geq0$} be a sequence of random variables defined on the probability space ($\Omega,\mathcal{F},P$). If $||.||_p$ is the $L^p$ norm defined as $||X_i||_p=(E[|X_i|^p])^{1/p}$, how should I ...
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0answers
25 views

Quick question on a naive-probability/counting exercise. [duplicate]

This is exercise 1.23 from Casella and Berger Statistical inference. Two people each toss a fair coin $n$ times. Find the probability that they will toss the same number of heads. My reasoning is ...
1
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2answers
39 views

Condition expectation exponential variable

Let $A=\{\Omega, \emptyset, [0,c],(c,\infty)\}$ be a sub-$\sigma$-algebra. I want to compute $E[X\mid A](\omega)$ for $\omega \in [0,c]$ and $X \sim \mathrm{Exp}(\lambda)$. Let's define $\mu(dx)=\...
0
votes
1answer
30 views

Exponential of Sums of different times of a Brownian Motion

Let $\{B_s\}_{s\in[0,1]}$ be a Brownian motion, let $t_1 < \dots < t_n \in [0,1]$, I am interested in finding good upper and lower bounds for $$ \mathbb{E}[\exp(B_{t_1}+ \dots + B_{t_n})]. $$ ...
0
votes
1answer
33 views

CDF and PDF of semaphore waiting time

Imagine we have a semaphore that alternates every 40 seconds between green and red. Waiting time is 0 when the semaphore is green, and when it is red it is the remaining time until it turns green. I ...
3
votes
4answers
403 views

How many ways in which distinct people can get off a train

I've been learning probability recently but I'm having trouble solving this question: Suppose you have 50 people on a train and you have 4 stations you can get off at (call them Stations 1,2,3,4). If ...
1
vote
2answers
29 views

Probability for a coin to be cut by exactly 5 lines

Assume you have a sheet of paper made of horizontal and vertical lines, and the distance between parallel lines is $8\ mm$. A circular coin with radius $1\ cm$ is being tossed on the paper. What is ...
2
votes
0answers
21 views

Showing that for a family of subsets of $[n]$ enough elements appear in high frequencies

Let $\mathcal{F} \subseteq 2^{[n]}$ a familiy of subsets. Assume that the following applies: For every $A \subseteq [n]$ , such that $|A|\leq \alpha n$ ($\alpha > 0$ is given), there's a subset $...
4
votes
0answers
65 views

Convex combination of independent random variables, that minimizes the $p$th moment

Suppose $V$ and $W$ are independent. Let \begin{align} f(x)=E[ ((1-x)V+xW )^p] \end{align} for $x \in [0,1]$ and some even $p \ge 2$. Find \begin{align} \min_{0 \le x \le 1} f(x) \end{align} ...
1
vote
1answer
37 views

Distribution of continuous distribution function of random variable

Suppose we have $X$ - random variable with distribution function $F(x)$, where $F(x)$ -- continuous distribution function. How to find the distribution function of new random variable $Y = F(X)$? ...
0
votes
0answers
28 views

Does there exist a collection of random variables satisfying given conditions?

Suppose $X,-Y$ and $Z$ are i.i.d random variables. I am trying to investigate whether there exists such random variables satisfying the following conditions: \begin{align} &a)\ X \rightarrow (X-...
-1
votes
1answer
32 views

Probability to get an $8$ on three $8$ sided dice, with $1$ die able to be rerolled [closed]

What is the probability to get an $8$ on three $8$ sided dice, with $1$ die able to be rerolled. Also, same but with $2$ dice and $4$ dice. Ty, I'm a bit rusty on my probability math.
3
votes
1answer
38 views

Solve probability equation

Given $T$ is t-distributed with $n=7$: $$P(T\geq -t) + P(T\geq 0)+P(T\geq t) + P(T\geq 2t) = 1.75$$ I did $$1-P(T\leq -t) + 1-P(T\leq0) + 1-P(T\leq t) + 1-P(T\leq 2t) = 1.75$$ $$P(T\leq 2t) + 1 + 0.5 ...
3
votes
0answers
55 views

What are some of good books in probability? [closed]

I am preparing for Ph.D entrances and my interest lies in probability and statistics. I am trying to learn it from William Feller's book but finding it quite difficult to comprehend. So please suggest ...
0
votes
1answer
54 views

Probability Rules proof

How to proof this? $$\mathbb{P}(A' \cup B') = [\mathbb{P}(A \cap B)']$$ because when I draw it, it does not look same. Or did I do it wrong?
0
votes
1answer
33 views

Probability of getting number greater than 4 given there is at least one tail .(Conditional Probability)

Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number ...
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0answers
46 views

Conditional probability of a product of random variables

Assume we have two instances $x_1$ and $x_2$ of the random variable $X$ that follow an exponential distribution with mean value 1. These instances are independent. Moreover, we have two instances $...
0
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0answers
15 views

Finding All Combinations of a Hierarchical list Where Conditions Are Involved

I want to find all possible combinations of a list that looks like this. a) Option 1 Sub Option 1 b) Option 2 Sub Option 2 c) Option 3 The catch is that there are some simple and some ...
2
votes
2answers
76 views

Mean of total time spent at the exhibition

An exhibition is open for a period of time of $T$ hours. Visitors arrive to it following a Poisson process with parameter $\lambda$ visitors per hour. Visitors stay in the exhibition until it's ...
0
votes
1answer
49 views

Probability of a coin toss six times resulting in three heads and three tails

What is the probability that a coin tossed six times results in three heads and three tails (in any order)? I am not sure how to count this using permutations and combinations. The complement does not ...
0
votes
2answers
59 views

What is the expected value of the sum of the $k$ (out of a set of $n$) smallest uniform random variables?

I know that the expected value of the sum of $n$ random variables is the sum of the expectation of each one. The expected value of a uniformly distributed random variable $U(a,b)$ is also well known ...
1
vote
2answers
42 views

Probability of getting destination with closed roads

So the question asks: We want to drive from A to B. See the road map in the figure below. Because of snow, each of the five roads (R1−R5) can be closed with probability $p$, independently of all other ...
3
votes
2answers
63 views

General strategies for evaluating sums in probabilistic/combinatoric problems: $\sum_n\sum_m {n+m \choose n}p^{n+m}(1-p)^{n+m}$

I have encountered the following summation: $$p^2\cdot \sum_{n=0}^\infty\sum_{m=0}^\infty {n+m \choose n}p^{n+m}(1-p)^{n+m}$$ This summation arises from naive analysis of this simple probability ...
1
vote
1answer
19 views

Finding whether function values are dependent or independent?

For the following question, I do not know what to do because I have only used dependence and independence on numbers, not functions. Two numbers are drawn at random from the set $ \{1,2,3,4\}$ If ...
0
votes
1answer
53 views

How to find this integral

I want to fin the following integral: $$\int_o^{x_n}\int_0^{x_{n-1}}\cdots\int_0^{x_2}n!\lambda^ne^{-\lambda(x_1+x_2+\cdots+x_n)}dx_1dx_A\cdots dx_{n-1}$$ I think that the answer is $\lambda ne^{-\...
1
vote
1answer
37 views

A box contains 4 piece of papers, each paper marked with A,B,C, and D respectively.

A box contains 4 piece of papers, each paper marked with A,B,C, and D respectively. A person draws a paper and observes its letter and puts it pack. Papers are now drawn repeatedly without ...
0
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0answers
35 views

If $\mathbb{E}\left[X \mid Y \right] = 0$ does $\mathbb{E}\left[X \mid Y,Z \right] = 0$

$X,Y,Z$ are all random variables on the same probability space. I think the title is false in general, but direction of a proof would be helpful. I also have some similar questions: $$ \text{If } \...
2
votes
3answers
49 views

Probability of a die ends up showing 1

So the question asks: A fair six-sided die is rolled repeatedly until one of the numbers 1, 2 or 3 shows up. Find the probability that the experiment ends with the die showing the number 1. So so ...
0
votes
1answer
30 views

Recognize elements of an hypothesis testing problem

Before the launch of a commercial product, a company makes a market survey to know the price that buyers are willing to pay. It is assumed that this price is normally distributed with a desviation ...
0
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3answers
30 views

What is the probability a piece of clothing was made by person 1 if it is defective

Person 1,2 and 3 produce the following proportions of clothes: Person 1: 10% Person 2: 30% Person 3: 60% The probability they make clothes that are defective are: Persion 1: 4% Person 2: 3% ...
0
votes
0answers
24 views

$I(X,Y,Z)$ and $I(X\bigcup Y,Z,W) \ge I(X,Y,W)$

I am trying to prove if $I(X,Y,Z)$ and $I(X\bigcup Y,Z,W)=> I(X,Y,W)$. I know that $I(X,Y|Z)=I(Y,X|Z)$ and $I(X,W|Z\bigcup Y)$ and $I(X,Y|Z) \Rightarrow I(X,Y\bigcup W|Z)$, unable to use the above ...
0
votes
1answer
17 views

Expected value of number of successes of n independent experiments

Assume n independent experiments with probabilities of $p_1,...,p_n$. Let's define the random variable as this: $$X= \begin{cases} \ 1, & \text{if exactly 1 of n experiments was successful} \\2,...
0
votes
1answer
70 views

Is it true that for any two events $A$ and $B$, $P(A)$ equals $0$ only if $P(A|B)$ is also $0$?

I know that the reverse is true, as $P(A|B) = (P(A)*P(B))/(P(B))$, so it holds that if $P(A)$ is $0$, then $P(A|B)$ must also be $0$, but can this statement be true?
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4answers
97 views

Problem about simple probability

I guess that this will be really simple for you guys, but i have no foundation in probability. Please, help me to find not only the answer but also what i need to learn in order to be able to solve ...