Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. 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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977 views

Probability with wins and losses

Suppose a series of games is played between two teams and that in any individual game, one of the teams wins and the other team loses. Let the probability that team A wins an individual game be p and ...
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Probability that dice gives 5, when we know for sure that outcome will be odd.

I am confused in this one, to solve it I used conditional probability formula but what should I have there in my numerator, if A is the event of 5 occurring and B is the event of always odd values. I ...
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What is the probability distribution of slots with x number of objects after m placements of n choose k permutations without repetition?

Problem I have 1000 slots. I first randomly place 20 balls in the 1000 slots. I can only place one ball per slot (no repetition). I then randomly place 40 additional balls in the same 1000 slots. I ...
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Question involving two Exponential random variables

Suppose that $𝑆$ and $𝑇$ are two independent random variables, with distribution $Exponential(\alpha)$ and $Exponential(\beta)$ respectively, for fixed and different $\alpha,\beta > 0$. What is ...
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Formual of selecting random elements from a generated pool of elements

Since I am not good at math and would like to know something, I ask here. I hope that this question can't be easy solved with Google, because I really have no idea how to call such a calculation to ...
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Find $E(X)$, where $X$ is the sum of consecutive successes in $n$ Bernoulli trails.

Let $X$ be the sum of consecutive successes in a $n$ Bernoulli trails with parameter $p$. for example, if $n=6$ and the results are $1,1,1,0,1,0$ then $X=2$. I want to find $E(X)$ for $n \ge 2$ and $...
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1answer
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Probability on birthday problem

I have a probability problem on birthday. Let's denote the 7 days in a week as Monday = 1, Tuesday = 2 ..., Sunday = 7. Pick N random people and the product of their corresponding birthdays' figures ...
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10 views

if X ~ Poi(λ) then calculate Expected value of Y = 2^X and Z = 1/(X+1)

if X ~ Poi(λ) then calculate Expected value of Y = 2^X and Z = 1/(X+1) I have no idea how to solve it. I tried Maclaurin series but it didn't work.
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What is the convergence rate of $\epsilon$ in strong law of large numbers?

Due to strong law of large numbers, we have for any $\epsilon>0$, there is n such that $Pr( \vert \overline{X}_n \vert -\mu <\epsilon)=1$. Could we possibly obtain for any $c>0$, $Pr( \vert \...
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1answer
17 views

Conditional probability problem with flights

Under good weather conditions, $80\%$ of flights arrive on time. During bad weather, only $50\%$ of flights arrive on time. Tomorrow, the chance of good weather is $60\%$. What is the probability that ...
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2answers
27 views

Probability to shoot a target

Three shooters aim at a target. The probability that they hit the target are $0.4$, $0.5$ and $0.7$, respectively. Find the probability that the target is hit exactly once. I don't know if this is ...
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Are random variables $X$ and $Y$ independent?

$f_{X,Y}(x,y)= \left\{\begin{matrix}\frac 27 (x+2y), \mbox{ } 0\leq x \leq1 , 1\leq y \leq 2 \\ 0, \mbox{ otherwise}\end{matrix}\right.$ My goal is to figure out if $X$ and $Y$ are independent. $f_{...
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Expectation of the sum of two Dirac deltas

Let $X_1,X_2$ be random vectors in $\mathbb{R}^n$ with joint PDF $f(x_1,x_2)$ such that $f(x_1,x_2)=f(x_2,x_1)\,\,\, \forall x_1,x_2 \in \mathbb{R}^{n}$ (i.e. $f$ is symmetric). Consider the following ...
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Calculating $E[X \mid X^2+Y^2]$ for normal and uniform distribution

Let's consider i.i.d random variables $X,Y$. I want to calculate $E[X|X^2+Y^2]$ in two cases : Assuming that $X,Y \sim N(0,1)$ Assuming that $X,Y \sim U(0,1)$ My work so far Thing which is very ...
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Randomly observing a process following two probability distributions one after the another

Consider a process that is evolving through a timeline, starting at $t=0$. The process can be described by two types of events (Type-I and Type-II) occurring one after the another (Type-I Type-II Type-...
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11 views

Regular Graph with a Partition Half of Neighbors Are in the Partition and Half Are Not

Hi Math Stachexchange, I am a Computer Science student who are new to this forum. I got troubled by this graph problem. Hearing that it is a useful forum, I wonder if I can get any helps here. To ...
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Generalized version of the Basic Counting Principle [closed]

Demonstrate the generalized version of the Basic Counting Principle. Hint: Use the Properties of the Cartesian Product of two sets A, B
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1answer
54 views

What is a conditional probability for 3 dice?

I roll three dice. Let's designate $A$, $B$, $C$ - the number of points on the first, second and third dice, respectively. I need to find the probability that points $A + B > C$ under condition $A &...
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11 views

Finding a set with probability satisfying an inequality, part 2 - slightly changed

Extended from Finding a set with probability satisfying an inequality, with additional assumptions added. I am requesting a hint, not necessarily a complete solution. The intuition behind this ...
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2answers
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Why is the conditional expectation failed case written this way?

This is my first post and I apologize in advance if I'm not using the right formatting/approach. Problem A coin, having probability $p$ of landing heads, is continually flipped until at least one head ...
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2answers
420 views

Biased coin flipped until $r$ heads appear

A coin having probability $p$ of coming up heads is successively flipped until the $r^{th}$ head appears. Let $X$ be the number of flips required. Find the probability $P(X=n)$. So this means I need ...
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13 views

Questions based on mean, standard deviation and sample size

We are given the following information (it doesn't matter what info it is but it is measured for a study of some sort): Mean:4.2 SD: 3.3 Sample Size: 234 Note: None of these questions should be using ...
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1answer
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Expected value of number of 3 red card sequences in a random shuffle of cards?

This was a question that I've seen before, and it came up in an interview and I couldn't get an answer. It seems like there is some reduction to make this problem simple but I couldn't find it. ...
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1answer
15 views

Show that $X_{n}\sim N(sin(πn),1)$ is uniformly tight

If $X_{n}\sim N(sin(πn),1)$, then how do I show $X_{n} = O_{p}(1)$? $X_{n} = O_{p}(1)$ means that for every $\epsilon$, there exists an $M$ such that $\sup_{n} P(|X_{n}|>M)<\epsilon$. I am ...
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2answers
22 views

Creating probability sample space of elementary events for a specific problem

I'm trying to create a sample space of elementary events for the following problem, however am having difficulty. Does anyone know how to create this set of elementary events? In a factory there are ...
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Binomial Distribution with Uniform Parameters [closed]

Researchers have found a COVID vaccine that causes harmful effects with probability $p$ where $p$ is uniformly distributed in the interval [0, 0.5]. To check the effectiveness of the vaccine, the ...
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1answer
29 views

An inequality in Erdős–Rényi random graph model

Let's consider Erdős–Rényi random graph model. This means that the probability assigned to a graph $G_{n,p}$ with $n$ vertexes and $m$ edges is equal to $$P(G_{n, p}) = p^{m}(1-p)^{N-m},$$ where $N = {...
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6 views

How to write and refer to the sample space of a categorical variable?

I have a variable with a sample space $\{A,B,C,D,E\}$ where $A<B<C<D<E$. I would like to define a variable $x$ that is a boolean variable where it is equal to $1$ when $x \geq D$. How ...
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1answer
40 views

Probability Struggle/Stick Problem

I am taking a probability class and I am struggling with this problem: Take a stick of unit length and break it into two pieces, choosing the break point at random. Now break the longer of two pieces ...
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15 views

Representation for the survival function of the multinomial distribution in terms of the Dirichlet density?

Let $K_p\sim \text{Binomial}(n,p)$ where $n\in \mathbb{N}$ and $0 \leq p \leq 1$. Simple computations show that, for $1 \leq k \leq n-1$ and $p_0\in (0,1)$, \begin{align} \mathbb{P}(K_{p_0} \geq k)...
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1answer
19 views

Given two indep standard normal RVs, X and Y, what is $P(X^2 + Y^2 \leq 1)$?

Given two standard normal random RVs, X and Y, how do you find $P(X^2 + Y^2 \leq 1)$? I approached this by integrating the bivariate using polar coordinates as $x^2 + y^2 = 1$. Not sure if this is ...
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28 views

What is the probability that any given patient has cardiovascular disease? [closed]

In the population that the physician's office serves, 29% have diabetes, 45% have cardiovascular disease, and 26% have COPD. 52% of those with diabetes, 45% of those with cardiovascular disease, and ...
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15 views

When to use combination and permutation

Sorry for the bad formatting below and its not really a good question title too, the real question may actually goes beyond what it is showing Lets go straight forward to 3 different probability ...
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2answers
26 views

How many integers between 1 and $10^4$ contain exactly one 8 and one 9? [duplicate]

question: How many integers between 1 and $10^4$ contain exactly one 8 and one 9? I try do this, but i don't know if this is good, i ask my teacher and she says this: IN A_3 more accounts look at ...
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0answers
9 views

How to Calculate the Output Multiplier of a Cumulative Chance Process?

I'm not sure what the proper way to describe what I'm looking for is, so apologies if the title is misleading. In developing a game, I am working on a particular algorithm that consists of multiple ...
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0answers
11 views

Does there exist a bounded function that preserves strict expectation inequalities?

Suppose that $\{X_i: i \in I\}$ is a collection of random variables on the measurable space $(\Omega, \mathcal F)$ taking values in the extended reals $[-\infty, \infty]$. Does there exist a ...
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1answer
25 views

Application of the central limit theorem on Markov chains

I'm studying a paper from Rosenfeld about bitcoin mining pools. He formulates a Markov chain by the definition: \begin{equation*} \begin{aligned} X_{t+1} - X_t= \begin{cases} B - (1 - f) \...
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2answers
36 views

N needed for probability > .001 of two people being a match at six genetic markers?

If for each marker there is a 1/9 chance that any two people are a match, how large would the sample need to be for the probability to exceed .001 that two people are a match at six markers?
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34 views

Breaking the password probability

An account uses 8-character passwords, consisting of letters (distinguishing between lower-case and capital letters) and digits. A spy program can check about 1 million passwords per second. a) On the ...
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Probability when choosing beads with repetitions

There are 20 different beads, 4 of which are yellow. Choosing 20 beads with repetitions, I need to calculate: The probability that exactly 5 of the chosen beads are yellow The probability that the ...
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18 views

first order moment of a random vector

I don't understand the notation \begin{equation}m_{\mathbf{Y},n}(y_1,\dots,y_n)= \int \langle y_1,y \rangle \dots\langle y_n,y\rangle f_\mathbf{Y}(y)\text{ d}y\end{equation} used in this paper (page ...
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Show that the continuous ranked probability score is strictly proper

I am having trouble verifying a claim from this paper (section 4.2). Let $\mathcal P$ be the set of all probability measure on the Borel algebra of $[0,1]$. The continuous ranked probability score for ...
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29 views

Prove that a conditional probability is a valid probability space

Let $(Ω, P(·))$ denote a probability space and suppose that B is an event in this space where $P(B) > 0$. Prove that $(Ω, P(·|B))$ is a valid probability space by showing that it satisfies the ...
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1answer
22 views

Data Processing Inequality for sufficient statistic case

Consider a Markov chain $ X \rightarrow Y \rightarrow Z $ and assume $Z$ is a sufficient statistic for $X$ (i.e $I(X;Y)=I(X;Z)$), do we have a case for $X, Y$ and $Z$ where $H(Y) > H(Z)$? Here is ...
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12 views

Conditional Probability Maths [closed]

Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. The company’s records indicate that the probabilities that good-, average-, and ...
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1answer
29 views

Finding the probability density function of a function of a continuous random variable

Let $$f_X(x)=\begin{cases}c \cdot x&\text{for }0 \leq x \leq 1\\ 0&\text{otherwise }\end{cases}$$ with $c > 0 $ be the probability density function of the random variable $X$. Find the ...
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1answer
24 views

Find conditional probability given the joint probability

Find the conditional distribution of $X$ and $Y$ given $Z = 0$ and the conditional distribution of $X$ and $Y$ given $Z = 1$ $P(X,Y|Z) = P(X, Y, Z) / P(Z) = P(X|Y,Z)P(Y|Z)$ So, For $Z = 0$: $f(x=0,y=...
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Can this be solved with the probability formula? [closed]

Can this be solved with the probability formula, and if not, then how can it be solved? In 24 hours time, two ships arrive independently into the harbor of Chewbakka Bay, denoted by A and B, ...
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12 views

What is the expected (average) range of H(+1), T(-1) with n trials?

Let n be the number of trials in a coin toss where each H is +1, T is -1, and we start at 0. Both H and T have a probability of .5. What is the average range of the distribution after n trials?
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How to solve this question of picking balls from bins, question is probability of ball number vs selection number?

I'm a student and always get confused with different combinations of balls in bins problem. I've encountered this question today and not able to solve this. Hope someone can give me some pointers, ...

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