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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6 views

What explains this difference in Poisson process' definition?

I am watching this video on Poisson process. The lecture video gives this definition (boxed in red): The book gives this definition (boxed in red): I would like to ask why $o(\tau)$ and $o_1(\tau)$ ...
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15 views

Expectation of a couple's date duration.

I encountered an question to the solution to a question in my probability textbook (Problem 4.23. Introduction to Probability 2nd edition, by Dimitri P. Bertsekas and John N. Tsitsiklis) The question ...
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18 views

Combination Problem on white table

Question : How many ways are there to choose 3 cells from a 4x4 table such that any two chosen cells do not belong to the same row nor the same column? What have i done so far : choosing $1$ from $...
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Expected strength of a list. [duplicate]

Consider a list of distinct positive integers from 1 to n. The strength of the list is defined as: the number of elements occuring in the list whose value is greater than all the previously occuring ...
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1answer
30 views

What is the meaning of difference in this question?

I was given the following problem: Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed $n$ times. What are the possible values of $...
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1answer
21 views

How many different options do you have for selecting 8 of these balls?

You have 6 red balls, 3 of which are identical & the other 3 are distinct & different from the previous 3, 4 distinct orange balls, 6 identical purple balls. You want to select 8 balls. How ...
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1answer
38 views

What theorem or concept should I use to approach this problem>

I have 4 objects in group A and 4 objects in group B. One by one I need to add the 8 objects from the two different groups in a line. But the condition is that as the line is being created, the ...
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1answer
18 views

Proof that a random variable is continuous

I am attempting to show that the below random variable has no atoms: $X = \sum_n \frac{\beta_{n}}{3^{n}}$, where $\beta_n$ is $0$ or $2$ with probability $\frac{1}{2}$.
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2answers
26 views

Prove that Random Variable is not absolutely continuous with respect to Lebesgue measure

The random variable is as follows: $X = \sum_n \frac{\beta_{n}}{3^{n}}$, where $\beta_n$ is $0$ or $2$ with probability $\frac{1}{2}$.
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1answer
20 views

Given the distance $d$ between two random points on a segment, find the MLE of its length $L$.

I could solve this problem, but I made a logic jump that I'm not sure how to justify. We have a segment of length $L$, and two random points with uniform distribution over that segment. That is, $p_{...
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20 views

Calculating probabilities of tornadoes

First of all, I am neither mathematician or statistician. I am an amateur who loves working with data. I have been trying to figure out to calculate the probabilities of tornadoes happening in certain ...
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$\sigma$-algebra generated by a random variable $X$ [closed]

On $\Omega=(-1,1)$ with Borel $\sigma$-field $\mathit{B}$, Let $X(\omega)=\omega^2$, which of the following subsets of $\Omega$ are in $\sigma(X)$: $A_{1}=(\frac{1}{4},\frac{1}{2})$, $A_{2}=\{0\}$, $...
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1answer
14 views

Is there a stochastic domination between “sub”Bernoulli random variables and Bernoulli ones?

Given an $n$-element set $[n]$. Let $R$ be a random subset of $[n]$, whose distribution is unknown, especially we do not assume independence in it. But what we know is that for all subsets $S\subseteq ...
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2answers
45 views

Finding the mean of the Poisson distribution using the moment generating function

Problem: Let $X$ have a Poisson distribution with parameters $\lambda$. Use moment generating function to find the mean and variance of $X$. Answer: For the Poisson distribution we have $M_x(t) = e^{\...
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1answer
23 views

Determine the distribution of the random variable $X_1+X_2$

$X_n= \begin{pmatrix} -1 & 0 & 1 \end{pmatrix}$ with $p = \begin{pmatrix} \frac{1}{2n} & 1-\frac{1}{n} & \frac{1}{2n} \end{pmatrix}$ is a discrete random variable. It's known that $...
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1answer
53 views

How to compute the expected minimum Hamming distance with 3 strings

If we sample $3$ binary strings of length $n$, uniformly and independently, what is the expected minimum Hamming distance between the closest pair? Numerically, it seems to be asymptotic to $n/2$ but ...
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1answer
16 views

Normal distribution, probability that the mean is in a confidence interval

We are given $X \sim \mathcal{N}(\mu, \sigma^2)$ , and $\mu, \sigma$ are unknown. The exercise is to find the probability that $\mu \in I_n$, where $I_n = \left[ \bar{X}_n - \frac{1.96}{\sqrt{n}}, \...
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15 views

Find the probability density of the variable x [closed]

Let X~N(5,(0.5)^2). Find the probability that after two independent trails, the random variable X in both of them will belong to the interval (2,4). Find the probability density of the variable X.
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24 views

Probability space-union of sets

Can someone help me with this problem? We let $(X,A,\lambda)$ be a probability space. We let $A_1,A_2,A_3,\ldots$ be a series of sets in $A$ which satisfy that $\mu(A_n)≥1-1/3^n$. Then we have to ...
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1answer
24 views

Example Conditional probability

Can You check pls ? Problem: there are 110 girls and 90 boys in the class,30% girls and 40% boys go to the school by bicycle.Probability ,that a boy gets into an accident is twice as much as a ...
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15 views

The distribution laws of x and y are given [closed]

The distribution laws of x and y are given below: X = -1, 0, 1 P = 0.3, 0.4, 0.3 Y = -1, 2 P =0.4, 0.6 supposing that X and Y are independent ,find the following characteristics: the distribution ...
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2answers
41 views

Weak and Strong Law of Large Numbers

Let $(X_n)_n$ be a sequence of independent random variables such that: $\mathbb P(X_n=1)=\mathbb P(X_n=-1)=\frac{1}{2}(1-2^{-n})$ $\mathbb P(X_n=2^{\frac{n}{2}})=\mathbb P(X_n=-2^{\frac{n}{2}})=2^{-...
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22 views

Extracting balls from an urn with balls of different colours and the r.v. is “different colours”

I have the following problem: I have $3$ balls from color $A$, $2$ from $B$, and $1$ from $C$. I want to find the elements of the muestral space of $X$: "different colours". I extract $2$ balls. So ...
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1answer
17 views

A biased dice is thrown 4 times

A biased dice for which the probability of getting the number 6 is $\frac{1}{6}$ , if it is thrown 4 times .Find the probability of getting at least one 6 ? My turn : The probability of getting one 6 ...
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36 views

Distribution of animals

I have R rabbits, F foxes and H houses in which I can put them (there is no information if R and F are greater than H or vice versa). How many methods of distributing them exist if: Every house has ...
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13 views

Number of mappings from A -> A with elements not mapped to themselves [duplicate]

For example the set $A = \{ 1,...,n \} $, how many mappings (bijections) would there be which ruled out any pairs $(1,1),(2,2),...,(n,n)$. I know that there are $n!$ total mappings.
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32 views

Expected maximum of a random vector

Is there an inequality of the following type: $\mathbb{E}[\max\limits_{j<k}\{x_j\}]-\max\limits_{j<k}\{\mathbb{E}[x_j]\}\leq C(k)\sum\limits_{j<k}\sqrt{Var(x_j)}.$ Where $C(k)$ is a ...
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3answers
27 views

Probability of forming a triangle

We are given a line with length $K$.We pick 2 random lines with length less than $K$.What is the probability to form a triangle? What I have done is : Let the lines be $x $ and $y$ , then we need to ...
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1answer
23 views

Bounding $\mathbb P[A_1,…,A_n]$ where $(A_i)$ are not independent

Given events $A_1,...,A_n$. We say that $i\leq n$ "is good" iff $A_i$ occurs and $i$ is bad iff $A_i^c$ occurs. We are given $\delta_1,...\delta_n\in[0,1]$ s.t. $\mathbb P[i \text { is good}]\geq1-\...
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2answers
26 views

Committee Selection Probability Question (AS-Level) [closed]

There are $15$ people from Swansea, $12$ from Wolverhampton and $10$ from Aberystwyth. A committee is to be made of at least $5$ people from Swansea and $2$ from anywhere else. What is the probability ...
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19 views

Is theoretical probability the mean of infinite empirical probabilities

I am a high schooler so pardon if the question is non sensical. If I run an experiments 'n' no. times, and organise them into groups of 100, will the mean of each group's empirical probability equal ...
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1answer
29 views

Two ways to calculate probability of pulling 2 cards out of a deck - why different results?

Let's say you wanted to know the probability of pulling out a Heart and a red card out of a standard shuffled deck of 52 cards, in that order. Option 1: 13/52 * 25/51 = 12.2% Option 2: (13 choose 1) *...
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1answer
23 views

Increasing event sequence question

In an exercise, if $ \{ X_n \} $ is a random variable sequence and $X$ another random variable, I am requested to show that the event sequence $\{T_n\}$ is increasing, where: $$ T_m = \{ |X_n-X| < ...
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2answers
36 views

Bayes theorem and application

A pregnancy test is accurate 97% of the time when someone is pregnant and 98% accurate when someone is not. Assuming that 60% of people who take the test are pregnant, and that someone tests positive ...
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29 views

Determine the probability that at least $70$ out of these $80$ random variables exceed zero.

Assume that $X_1, X_2,\ldots X_{80}$ are independent and identically distributed with the same distribution as $X$ (hence $X_i \sim F(x)$ for all $i=1,\ldots,80$). Determine the probability that at ...
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1answer
16 views

Markov chain - expected times to absorption

The expected times to absorption for Markov chain is calculated from the following equations: I would like to ask why 1 is there in the second equation. Thanks.
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1answer
21 views

Show that the smallest sigma field of the subsets of $A$ containing the sets open in $A$. [closed]

If $A$ be a Borel subset of $\mathbb{R}$,show that the smallest sigma field of the subsets of $A$ containing the sets open in $A$ is $\{B \in B_{\mathbb{R}}: B \subset A\}$ . I was taking $C = \{\...
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18 views

Binomial Random Walk Motion

I am looking at the following binomial random walk: Let $S(t)$ be the underlying price at time $t$, and we assume that at the next time step, the possible options for $S(t+\delta t)$ are $uS$ with ...
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1answer
18 views

A generalisation of the Brier score that takes account of the relative spacing of outcomes?

The Brier score is a measure of the accuracy of a set of probabilistic predictions, where each prediction is considered against an output function that takes the value either 0 or 1. There are ...
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32 views

Question about a uniformly distributed stochastic process

Q) Let rain drops of radius $1$cm fall uniformly on a surface and the drops may overlap. What density, in drops per square meter, of rain drops does it take to cover $90\%$ of the surface. Covering $...
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1answer
50 views

Probability no male- female pairs share same birthday

Lets say there are 8 people in a room. There are 4 males(M) and 4 females(F). What is the probability that there are no M-F pairs that have the same birthday ? It is OK for males to share a birthday ...
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1answer
65 views

Probability of choosing integers that sum to zero

This is just a little problem that occurred to me. I don't know how to go about solving it and thought it would be fun to post here. Let $N \geq 3$ be a natural number, and let $n \leq N$. What is ...
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33 views

Monte Carlo Simulations [closed]

[Part a) Let $X ∼ N (0, σ^2)$ and suppose you use a standard Monte Carlo simulation to estimate $θ = E[(X − K)+]$. When the number of samples used is large, what fraction of the total samples will ...
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2answers
24 views

Finding probability of a standard deck of 52 French (Bridge) cards

Need some help here, I am confused here. I got 2. Probability = (1/36)*(1/4^4) = 0.000109, but my prof told me that I am wrong. Any help will be greatly appreciated.
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1answer
35 views

Prove $\lim_{\varepsilon\to 0^+}\frac{1}{\varepsilon}\int\limits_{X\leqslant \varepsilon}X\mathrm{d}\mathbb{P}=0$

Let $X$ be a random variable taking values in $[0,+\infty]$. Prove that: $$\lim_{\varepsilon\to 0^+}\frac{1}{\varepsilon}\int\limits_{X\leqslant \varepsilon}X\mathrm{d}\mathbb{P}=0 ~~~~\mathrm{and~~~...
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1answer
37 views

Would computer keyboard be an example of a one to one function?

Would a computer keyboard be an example of a one to one function? For example, a user inputs A and the output would be A, which appears to be a one to one function to me. However, in the event that ...
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0answers
18 views

Borel algebra in probability theory. [closed]

What is Borel algebra in probability theory and why it's important? Basically I want to know why would I need Borel algebra in probability theory in a layman's term(UG). Also, why is the Borel algebra ...
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1answer
25 views

Finding $E[Z]$ and $P\left(\sqrt Z\le \frac32\right)$ where $Z=\min(X_1,X_2)$

Let $X_{1},X_{2} $ be independent and $X_{1},X_{2} \sim U(1,6)$, $Z= \min(X_{1},X_{2})$, $ T = \sqrt{Z}$. Find $EZ$ and $P(T \le \frac{3}{2})$. What I tried is the following : I took a geometric ...
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29 views

What is the probability that a > b+c if a, b and c are three independent random variables uniformly distributed over segment [0,1] [closed]

Let $a, b, c$ be three independent random variables uniformly distributed in the segment $[0, 1] $. What is the probability that $a > b + c$? 0.5 0.333 0.25 0.1666 0.08333
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10 views

Probability Given Three Ranges for Variables in an Equation [duplicate]

I learned that the equation \begin{align*} |\textbf{F}| = \frac{Gm_{1}m_{2}}{r^{2}}. \end{align*} Is used to solve for the force of gravity between two different masses ($G$ = newton's gravitational ...