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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Probability $a+b>c$ for $a,b,c$ uniform in $[0,1]$

Let $a,b,c$ be drawn independently and uniformly at random from $[0,1]$. What is the probability that $a+b>c$? We can represent $(a,b,c)$ as points in a unit cube, but as we want to consider the ...
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Statistics Question about probability of twins

23 out of 1000 births result in fraternal twins, 4 out of 1000 in identical twins. Identical twins must be the same sex, but the sexes of fraternal twins are independent. For simplicity we assume ...
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1answer
15 views

Prove that the probability does not depend on the number of elements randomly removed

Suppose you initially have $N$ balls in a barrel, $n$ of which are black and $N-n$ of which are white. Now suppose $k$ balls are randomly removed from the barrel, and you don't get to see what color ...
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Can anyone help with this probability question?

n balls are arranged in n boxes (the balls are distinguishable and each box can accommodate any number of balls). What is the probability that one box stays empty? The answer is $$n!(n-1)\over ...
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11 views

Fair coin is tossed until head is obtained, what is probability having exactly same tosses

Consider an experiment in which a fair coin is tossed until a head is obtained for the first time. If this experiment is performed three times, what is the probability that exactly the same number of ...
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13 views

Joint Probability Density Function of a Sample of A Normal Distribution

Here is the question: Suppose that a random variable is normally distributed with mean μ and variance σ^2 , and we draw a random sample of five observations from this distribution. What is the joint ...
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7 views

Expectation of vector normalization

I have problem like this $\theta$ is from Dirichlet distribution $\theta \sim Dir(\alpha)$ Assume that $\theta$ is the vector of $K$ elements and I want to calculate $$ \mathbb{E} ...
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8 views

Finding Lifetime Incidence Based on Annual Incidence

This question should be simple, but I can't wrap my head around it. The thing I can't wrap my head around most of all is how to calculate lifetime incidence among a whole population based on a certain ...
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0answers
16 views

Linearity of expectation for infinite sums?

I have a question related to this post: Expected value of infinite sum Is the condition listed necessary/sufficient (or both?) For instance, I'm thinking of $X_n=\frac{1}{n}Z_n$, where $Z_n ...
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14 views

Markov Inequality for almost positive random variables

I came across the following problem in my research, where I want to apply Markov inequality to bound the tail probability of a random variable, X. However, the random variable X is not strictly ...
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25 views

Using the total probability

If a coin is tossed repeatedly and the outcomes are recorded as a sequence of H’s (heads) and T’s (tails). Define each set of consecutive H’s and each set of consecutive T’s as a run, e.g. the ...
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Is this formula for disintegration correct?

I struggle to understand the concepts of 'regular conditional distributions' and disintegration. In this case, let $(X_t : t \geq 0)$ be a stochastic process (let's say it takes real values, for ...
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3answers
22 views

Conditional Probability : Method to approaching problems?

Or at least i think these problems are conditional probability. Im having trouble approaching a probability problem in general. Here is one problem that confused me. ---A survey of a magazine’s ...
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1answer
20 views

Conditional probability for 3 variables

I am reading this document. On page 22 they have given this figure and then asked P(tea|88005 & organic)? The formula they have used to find it is P(tea|88005 & organic) = P(88005 | tea) ...
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9 views

How do you normalise in a sequential bayesian aproach

Lets say you have an event G that a defendant is guilty, and you have variable F probability of finding his footprint at a crime scene assume you have the following information beforehand: so you ...
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38 views

Is p(x)dx equal to dp(x)?

I'm confused with the definition of the expectation operator. Assume a random variable $X$ having a probability distribution $p(x)$. Then the expected value of $X$ can be computed as $\int xp(x)dx$. ...
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23 views

Hypergeometic Distribution

How does the hypergeometric distribution: $\Pr[X = k] = \frac{\dbinom{m}{k}\!\!{\dbinom{N-m}{n-k}}}{\dbinom{N}{n}}$ deal with the fact that the objects being sampled are taken without replacement?
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1answer
30 views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
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2answers
32 views

A box contains 3 red, 8 yellow, and 13 green balls

A box contains 3 red, 8 yellow, and 13 green balls. Another box contains 5 red, 7 yellow, and 6 green balls. One ball is selected at random from each box. What is the probability that both balls will ...
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0answers
10 views

Calculating error percentage for rolling dice

I'm trying to figure out how an error percentage is calculated when rolling 2 6-sided dice. The table I am looking at shows the sum of the dice, the actual number of times a sum was rolled, the odds, ...
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1answer
39 views

How spicy are my peppers?

This came naturally to me when I studied statistics ~10 years ago, but for the life of me I can't remember how to work this out: For a given chilli pepper, there is (let's say exactly) $1/100$ ...
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1answer
12 views

Probability of dice throw

Suppose that 8 dice are rolled. What is the probability that the sum of the eight dice is 9? I would interpret this question as: What is the probability that we get exactly 7 ones and one 2. We have ...
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1answer
35 views

probability that two people share the birthday and the date of death

I've encountered an interesting probability problem that my little amount of knowledge does not help me to solve. Select 1000 people from the past, and ignore the year. Also assume that the 365 days ...
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0answers
11 views

Aggregate Loss Model (Actuarial Science)

Question: Let $S$ be the random sum of $N$ independent frequencies {${M_i}$} where $N$ is negative binomial distributed with $r = 5$ and $\beta= 3$ and ${M_i}$ is Poisson distributed with = 3. Find ...
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19 views

Poisson Process and Bernoulli Trials

Fix λ>0. Let $f_{n,λ/n}$ be the PMF of the binomial distribution with parameters n and λ/n. I have calculated that: for k=0, $$\lim_{n \rightarrow \infty} f_{n,λ/n} = e^{-λ}$$ for k>0, $$\lim_{n ...
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0answers
21 views

Is it possible to calculate the balanced cost of parameters' increase in card game? How?

I wonder if it's possible to calculate the balanced cost of parameters' increase for the card game. Game rules: Each player draw 7 cards at the beginning of the game and then one card each turn ...
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16 views

The expected number of polygons created as a result of the intersection between randomly placed rectangles inside a square

How can I compute the expected number of polygons created as a result of the intersection of $k$ rectangles of area $B$ each, which fall randomly inside a square of area A? Regarding how randomness ...
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0answers
17 views

evaluating a probability on a multivariate normal

Let $(X_1,X_2,X_3,X_4)$ be zero mean jointly normal with covariance $\Sigma$. I would then be thankful if you help me with hints as how to evaluate $$P(X_1^4X_2^2-X_3^4 X_4^2\leq 0).$$ This should ...
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1answer
29 views

Volume and Probability of a region given by a random variable

I am currently reading this paper. It is about nearest neighbors of a query point $X_q\in\mathbb{R}^k$ within a point set $P=\{X_i\mid X_i\in\mathbb{R}^k\}$, where the points have distribution $p(X)$ ...
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1answer
44 views

The expected area of the intersection of randomly placed rectangles [on hold]

How can I compute the expected area of the intersection of k rectangles of area B each, which fall randomly inside a square of area A? Regarding how randomness is applied, first the centre of the ...
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0answers
10 views

Special case of bernoulli's trail

Lets assume that there are $n$ number of trains visiting $m$ stations. What is the probability that $k$ out of $n$ trains meet at $j_{th}$ station given that $P_{ij}$ is the probability of $i_{th}$ ...
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0answers
15 views

Poker dice probability of rolling 2 pairs

Poker dice are played by rolling 5 dice. Let A be the event of rolling 2 pairs. (e.g. 1,1,2,2,3.). Find $\mathbb{P}(A)$. So my answer is as follows: $$\mathbb{P}(A) = ...
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20 views

The probability of the element at index $i$ is greater than all following items

Consider an array which is a permutation of $[1,2,3,\dotsc,n]$. Each of the permutation is equally likely (Uniform distribution). What is the probability that the item at index $i$ is greater than ...
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1answer
34 views

2 identical candies are given to 2 distinct children at random. What is the Probability that each children has at least one candy?

OK i have two different ways to approach the problem and both give different answers, it would be great if someone could clarify which of them is correct(if they are). Method 1: Since the candies are ...
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0answers
12 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other ...
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2answers
21 views

Conditional probability rolling two dice

Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? I already know the answer, but am having some trouble ...
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0answers
15 views

Linear conditional characteristic function

If $X$ is an $\mathbb R ^n$-valued random vector with characteristic function $\phi _X(k)$, $a$ and $b$ are non-random matrices, and $c$ is a non-random vector, then what is the characteristic ...
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2answers
20 views

Equivalence involving expectation

I am stuck with the following problem, where I am asked to prove/disprove the following hypothesis: Is $\mathrm{E}\{e^{\max_i X_i}\} = \mathrm{E}\{\max\limits_i e^{X_i}\}$, where the $X_i$'s are ...
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24 views

Probability of rolling one die by two persons 3 times each [on hold]

So suppose you have one die. Person A rolls one die 3 times. Person B does the same after A is finished. What is the probability that person A's smallest number (of his/hers 3 rolls) is greater than ...
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1answer
35 views

What's the probability that 2 of 4 people were born in the same month

I have this problem to solve: "Calculate what's the probability that in the four-person family at least 2 people were born in the same month". I know that I can calculate opposite event in very ...
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1answer
22 views

Expected value for random walk

A point starts at the origin and can randomly go up, down, left, right (equally likely). The question asks to write the expression of the point's position in terms of $x_1$ -units up, $x_2$ -units ...
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3answers
61 views

A box contains 10 balls numbered from 1 to 10

An urn contains balls numbered $1, \ldots, 10$. Five balls are drawn without replacement. What is the probability that the second largest of the five numbers drawn will be 8? I believe the number of ...
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1answer
6 views

formula to produce a set of probability distributions for a set of integers between a lower and upper bound with a given mean value

The goal is to establish a set of probabilities to be used to select an integer value where the probability of selecting I is Q, I+1 is R, I+2 is S, ... I+n is Z and such that the integer with the ...
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2answers
24 views

No pairs when drawing cards from deck

Suppose we are dealt five cards from an ordinary 52-card deck. What is the probability that we get no pairs (i.e. all cards are different values). I'm not sure if I've got the right answer on this ...
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28 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
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2answers
39 views

Probability involving cards

Given a deck of 8 cards containing 4 jacks and 4 queens, if 4 cards are selected with replacement (I put the card back after I pull it) how can I calculate the probability that no consecutive cards ...
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0answers
10 views

Confusion over Multivariate Hypergeometric Distribution

I'm reading Freund's statistics book and he says the following: "Just as the hypergeometric distribution takes the place of the binomial distribution for sampling without replacement, there also ...
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1answer
26 views

An inequality involving the normal distribution

Let $X$ be normal random variable with parameters $\mu = 0$ and $\sigma^2 =1 $. Is it true that $$ P( X > x ) \leq \frac{1}{x \sqrt{2 \pi}} e^{ -x^2/2} \text{ for }x>0 \text{ ??}$$
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Seemingly dependent problem broken down to independent events?

The problem is: A box contains 18 tennis balls, of which 8 are new. Suppose that three balls are selected randomly,played with, and after play they are returned to the box (if they were new, they are ...
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1answer
26 views

Distribution of maximum of correlated Gaussians

Let $X_1,X_2,...,X_n$ be iid standard Gaussian random variables. Consider the set of random variables $M =\left\{\left( X_i-X_j\right) :i,j = \left\{1,2,\dots,n\right\} \& i\ne j\right\}$. I ...