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Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

2
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2answers
96 views

Apex angle of a triangle as a random variable

I am not an expert in Probability Theory and I apologize if I make some mistakes in "translating" the following problem into the language of random variables. Any help also to improve the formulation ...
1
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2answers
61 views

In how many ways can we split $80$ persons in a $5$ wagon train such that :

In how many ways can we distribute $80$ persons in a $5$ wagon train such that : $a)$ exactly $15$ go into the first wagon $b)$ exactly $15$ go into one wagon For $a$) we have $\binom{80}{...
0
votes
1answer
28 views

Chance of finding cattle in streams

Last year I went fishing and crossed 4 farms to fish 4 different rivers. On 3 occasions I saw cattle in the river. Given there are 15,000 dairy farms in New Zealand I would like to understand the ...
0
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2answers
31 views

Sum of #s on dice.

Six standard six-sided dice are rolled, and the sum $S$ is calculated. What is the probability that $S × (42 – S ) < 297?$ Express your answer as a common fraction. First off can I ONLY just have ...
-2
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1answer
31 views

conditional probability 5 [on hold]

If P(A)=0.61 P(B)=0.56 and P(A U B)=0.81, calculate the following. ①P( A / A ∩B) The answer is 61/81. ➁P( A /A'∩B) The answer is 13/25. Please teach me how to solve these.
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0answers
42 views

Expected value of symmetric random variables

A random variable $X$ is symmetric if $X$ has the same probability distribution as $-X$. In the discrete case symmetry means that $P(X = k) = P(X = -k)$ for all possible values of $k$. In the ...
0
votes
0answers
25 views

Moment List of Exponential [on hold]

I need to follow this outline to find the moment list of Y ~ Exp(𝜆), 𝜆 = 1/E(Y) > 0. Let X ~ Exp(1). Show that X/𝜆 ~ Exp(𝜆), by finding the cdf of Y and then differentiating it. Write, L(t) = ∫∞ �...
1
vote
0answers
21 views

Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family? Details: Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\...
0
votes
0answers
37 views

Reflected Brownian motion on a compact interval whose lenght is going to 0

Suppose I have a Brownian motion $X_t$ in the interval $[0, \gamma]$ which is reflected at the boundaries $0$ and $\gamma$. We call $P_\gamma$ the law of such reflected Brownian motion. The generator ...
0
votes
3answers
64 views

Is “30 times more likely” equivalent to a “3000% greater probability”?

I am trying to make a persuasive point based on facts and would like to be most accurate / clear in my point. Would "30 times more likely" be equivalent to a 3000% greater probability"
0
votes
3answers
48 views

Almost sure convergence of the sum of a series of non-identical random variables

Suppose we have a sequence of r.v.s $Y_n$ for $n\geq 0$. Suppose $\mathbb{P}(Y_n=1)=\mathbb{P}(Y_n=-1)=\frac{1}{2^{n+1}}$ and $\mathbb{P}(Y_n=0)=1-\frac{1}{2^{n}}$. We have $X_n=\sum_{t=1}^n (\frac{1}{...
1
vote
0answers
34 views

Continuous random variables with zero variance

I have a very simple question. I heard in my class that a random variable is equal to zero if its variance is equal to zero. I understand that if the variance of a random variable is zero, then that ...
0
votes
1answer
29 views

What is the expected time to get out of the Maze?

There are three doors in a Maze. Choosing each door equal probability. If we choose the $1^{st}$ door, then after 5 minutes, we can go out of the Maze. If we choose the $2^{nd}$ door, then after 10 ...
0
votes
2answers
22 views

What is the probability of getting a certain combination of marbles in $3$ draws? [on hold]

A bag contains 3 red marbles, 4 white marbles and 3 black marbles. Find the probability of getting a red marble on the first draw, a black marble on the second draw, and a white marble on the third ...
0
votes
1answer
39 views

Recompute expected value of the sum of value of five face down cards when the lowest value card is replaced by a new face down card

Recently I heard of this puzzle/riddle type question that was asked in an application interview and I am unable to solve it. The problem is as follows Stage 1: The interviewer holds a regular deck ...
3
votes
2answers
78 views

Monty Hall with 4 Doors Solution

I am trying to analyze a Monty Hall question with four doors (3 goats, 1 car) just so I can then apply the problem with n doors. I applied Bayes' theorem, calculated the probabilities and am trying ...
0
votes
2answers
40 views

A family has two children. Given that one of the children is a boy, what is the probability that both children are boys? [duplicate]

A family has two children. Given that one of the children is a boy, what is the probability that both children are boys? I was doing this question using conditional probability formula. Suppose, (1)...
0
votes
1answer
16 views

derivation of softmax

This is the derivation of softmax in Bishop's PRML: $$ln(\frac{u_k}{1-\sum_ju_j}) = n_k$$ "Which we can solve for $u_k$ by first summing both sides over k and then rearranging and back-substituting ...
0
votes
1answer
42 views

Modification of the law of large numbers for Binomial random variables. [duplicate]

Let $(p_n)_{n \geq 1}$ be a sequence of numbers in $[0,1]$ such that $p_n \to p$. Let $(X_n)_{n \geq 1}$ be a sequence of independent random variables where $X_n \sim Bin(n,p_n)$. Is it true that $X_n/...
4
votes
1answer
81 views

What is the distribution of $(1+X^2)e^{-X^2/2}$ when $X$ is Cauchy?

If $X$ is a Cauchy random variable with $f(x)=\frac{1}{\pi}\frac{1}{1+x^{2}}$, what is the distribution of $Y= (1+X^2)e^{-X^2/2}$ ? What I tried: I was thinking I may be able to use Jacobian, but I ...
0
votes
2answers
37 views

probability question on casting a pair of dice

A pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show that the probability of the six and eight occurring before two sevens is 0.546. An ...
-1
votes
0answers
31 views

Basic concept of Probability

1.A box contain 10 pairs shoes, if 8 shoes are randomly selected, what is the probability that there will be a. No complete pair b. Exactly one complete pair 2.Twi die are are used, each loaded ...
0
votes
1answer
13 views

Set theoretic notation for statistical sampling without replacement?

I would like to know how to use set theoretic notation to define a subset $D$ of a finite event space $S$ that contains exactly $n$ elements $d_i$ which are sampled uniformly without replacement from $...
0
votes
0answers
51 views

Probability of a straight in a card game I'm designing

I am developing a card game with a friend and I want to calculate the probability of different win conditions being achieved (I am putting together an excel sheet). I have trouble calculating the ...
0
votes
2answers
25 views

Rearrangement problem of the letters of a given word

In how many ways can we rearrange the letters in the word INDIVISIBILITY such that no two 'I's are adjacent to each other? My try : Total number of rearrangement is ${14 \choose 6} ×8! $.I'm trying ...
2
votes
2answers
37 views

Expected value of number of throws of a dice to get element $1$ four times

Here is the question: Find the expected value and variance of the number of times one must throw a dice until the element $1$ has been obtained $4$ times. My attempt: The minimum number of throws ...
0
votes
1answer
24 views

How do I prove this for a borel sigma algebra on R?

I know that I much show each side is a subset of the other, but i am not sure where to start.
1
vote
1answer
36 views

Slick way to solve P(pattern A appears before pattern B) type of problems in a coin toss process

Let's consider a fair coin and an infinite toss game. We frequently got asked about how to compute the probability of pattern $A$ appearing before pattern $B$ where $A$ and $B$ are both one of the ...
1
vote
1answer
41 views

What is $F_\mu(\mu)$?

Given a measure $\mu$ and the corresponding distribution function $F_\mu$. What happens if one looks at $$F_\mu(\mu)~? $$ One might as well assume that $\mu <<\lambda$. Thanks in advance!
2
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0answers
74 views

Unbiased real vector with respect to arbitrary orthonormal basis for a finite Hilbert space

Here I use Dirac notation to denote vectors. I would like to show that for an arbitrary orthonormal basis $\{ |\psi_k\rangle \}_{k=1}^n \subset \mathbb C^n$, $$\langle \psi_i | \psi_j \rangle = \begin{...
-2
votes
1answer
42 views

Combinations with Piano Keys

How many sound combinations can be created by the $10$ selected piano keys if each sound combination contains from $3$ to $10$ keys? Can anyone throw at least a hint? I am having difficulties with ...
0
votes
1answer
28 views

Does $P(\liminf_{n \to \infty}\{|X_{n}|\leq \epsilon\})=1\iff \exists N \in \mathbb N, |X_{n}|\leq \epsilon, \forall n \geq N, P-$a.s.

Background to my question: Given that $(X_{n})_{n}$ are random variables on $(\Omega, \mathcal{F}, P)$ and for $\epsilon > 0$: $\sum_{n \in \mathbb N}P(|X_{n}|>\epsilon)<\infty$ It follows ...
0
votes
1answer
33 views

Properties of stochastic ordering

On the entry for stochastic ordering in Wikipedia it is stated that If $A\preceq B$ and ${\displaystyle {\rm {E}}[A]={\rm {E}}[B]}$ then ${\displaystyle A{\overset {d}{=}}B}$ (the random variables ...
2
votes
2answers
24 views

Equivalence of the sum of random variables and their expectation

Given that $X$ is a random variable I define $$ \psi = \sum_i^{n} X_i $$ so $\psi$ is the sum of $n$ variables with the same distribution. Given that $$ \bar{X} = \frac{\sum_{i}^{n} X_i}{n} $$ I ...
0
votes
1answer
14 views

Is the use of GLS appropriate in case of statistically independent errors in linear regression?

Let $Y_i = \beta x_i + e_i $, where $e_1 ~ N(0, \sigma^2)$ and $e_2 ~ N(0, 2\sigma^2)$, and $e_1$ and $e_2$ are statistically independent. If $x_1 = 1$ and $x_2 = -1$ obtain the weighted least squares ...
0
votes
1answer
17 views

Computing covariance and variance of a random variable

I toss a fair coin three times. Let $X$ denote the number of heads and let $Y$ denote the number of tails. (a) Find $\text{Cov}(X, Y)$ (b) Find $\rho(X, Y)$ (a) $\text{Cov}(X, Y) = E(XY)...
0
votes
1answer
12 views

Question regarding expected value of “composition”

The question is to calculate $Emin[X_1,...X_{N+1}]$, where $N,X_1,...$ are independent, and $N$ has poisson distribution with parameter 1, and $X_i$ has expotential distribution with parameter 2. My ...
0
votes
1answer
70 views

The probability that a candidate comes with all $3$ pens having the same color [duplicate]

Question Candidates were asked to come to an interview with $3$ pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a ...
4
votes
2answers
69 views

Bounded 3 values random walk

I have an array of size M, for each shift one random variable $X_i$ enter and one exit. The $X_i$ r.v. are iid and $X_i = \pm1$ with $p=\frac{1}{2}$. Assuming that the sum of the $M$ $X_i$ random ...
0
votes
0answers
21 views

Joint Bernoulli distribution given marginal distributions and correlation

General Info: I am in a situation where I have the marginal distributions for some number of Bernoulli random variables and a fixed correlation between each. I need to know the joint distribution of ...
4
votes
2answers
539 views

King on reduced chessboard $2\times 2$ moving randomly, what is the probability that it ends up in one of the corners after $1000$ moves?

As mentioned in the title, we have a chessboard $2\times2$, the king moves with equal probability to each square on the chessboard. King begins from the left upper corner. What is the approximate ...
0
votes
0answers
15 views

Expected value of falling factorials from axioms of Poisson process

Falling factorial, $(x)_n$, is the product of biggest $n$ terms in factorial, $(x)_n = x(x-1)(x-1)\cdot \ldots \cdot (x-n+1)$. Or the number of ways to color the set of $n$ objects into different ...
0
votes
1answer
21 views

Is my proof valid? Let $\{A_i\}_{i=0}^{i=n}$ a series of events such that $\forall i$ $P(A_i)=1$. Show that $\bigcap\limits_{0 \leq i \leq n}A_i=1$.

Let $\{A_i\}_{i=0}^{i=n}$ a series of events such that $\forall i$ $P(A_i)=1$. Show that $\bigcap\limits_{0 \leq i \leq n}A_i=1$. My attempt: Let $0\leq k\ne j\leq n$, so $P(A_j)=1, P(A_k)=1$. ...
1
vote
1answer
61 views

Typo in Ross' Introduction to Probability Models 11th ed p. 205?

possible typo I have problems understanding the text marked in red. Is this supposed to say something else? The syntax just seems off to me.
3
votes
0answers
72 views

A peculiarity of Eisenstein polynomials

I've tested polynomials of degree 6 with random integer coefficients $|a_i|<50$ in test-series of $10,000$. The probability of a random primitive polynomial of the kind to be reducible seems to be ...
1
vote
1answer
26 views

Double conditional in a probability

Is it is permissible to have a double conditional in probability? I've searched google and this site but couldn't find anything like the following: $P(A | B | C)=?$ If it is legit, would it be ...
1
vote
1answer
22 views

Probability density function on uniform distribution

Let $R=\{(x_1,x_2) \in \mathbb{R}^2 ~|~ x_1^2 + x_2^2 \leq 1\}$ be the unit ball in $\mathbb{R}^2$. It was stated to me that $\rho$ is the uniform probability on the square $[-1,1]^2$ and that $\...
3
votes
2answers
50 views

Probability of getting 3 balls in 1st box if 12 balls are distributed randomly among 3 boxes

$12$ balls are distributed at random among $3$ boxes.The probability that the 1st box will contain $3$ balls is_______? My Approach: $\quad \quad \quad \quad \quad \quad \text{Method}1$[Considering ...
2
votes
0answers
61 views

Showing inequality of probabilty generating functions with the mean value theorem

Let $Z_n$ be a nonnegative discrete random variable (Galton-Watson Process in this case). I need to show that: $$\frac{f(\phi_{n}(\frac{u}{m}))-f(\phi_{n-1}(\frac{u}{m}))}{u} \le \frac{\phi_{n}(\frac{...
0
votes
1answer
25 views

Statistics Independence Question: Given that an item has passed inspection, what is the probability that it is actually flawed?

The problem I have a question about is below. I only have a question on part (e), but I included the other parts of the question and answers as a reference. A quality control inspector is examining ...