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.

67,666 questions
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Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value

Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $c$? I've messed ...
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Probabilty Combinatorial related; The Messy Mail Man

Given the messy mail man problem of n pieces of mail, and X RV which value is "how many pieces of mail arrived to the right mail box," how do I compute E(X)? I saw a solution using indicators, but I ...
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The Nature of Probability Mass/Density Functions

Consider a certain random variable and all its possible probability mass functions (or probability density functions). What structure does this space have? For example, it can be endowed with a ...
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Interpretation of a probability problem: expected value.

I am having a few doubt on the interpretation of this problem that I have read on book about interviews questions. Here the text: A mythical city contains N=100,000 married couples but no children. ...
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Distribution of a continuous random variable

I was reading through my"Random Variability in Business Situations" notes and wanted to enquire about some difficulty I've encountered. On the fourth and final column "probability calculated from ...
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Mean excess life conditionally on current life in a renewal process

I am self-studying renewal processes and I came across an interesting problem. How would one go about finding the expectation of excess life given that current life is equal to x? I am assuming in ...
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MLE of parameter in distribution of gaussian time series [closed]

suppose $X_1,X_2,\ldots,X_n$ have joint distribution. if $X_1\sim\mathcal{N}(0,1)$ and for $j=0,1,\ldots,n-1$ conditional distribution $X_{j+1}|X_1=x_1,\ldots,X_j=x_j\sim\mathcal{N}(\rho x_j,1)$, then ...
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Probability Combinatorial related; rolling dice

when 3 fair dices roll, let Y be a Random Variable of which value is the minimum result of all 3 dices results. for example if the results are (2,3,2) then Y=2. how do i find, in the most elegant ...
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Rolling dice, fixed number of trials, when to stop rolling?

Suppose you have one $k$-sided die (labeled $1, \ldots, k$). You want to roll the highest number you can. You get $T$ trials. When you roll, you can either accept the number you rolled, or you can ...
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100 prisoners and a lightbulb

100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb ...
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Strategy in an urn problem with two urns

We consider two urns each containing $N$ balls. One of the urns has white balls only. The other has $k$ black balls and thus $N-k$ white balls. Both $N$ and $k$ are known, and $k>0$. What would be ...
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Probablity of picking up 3 vowels from the word “MATHEMATICS”

Each of the letters in the word "MATHEMATICS" is on a letter tile in a bag. Foool picks three without replacement. what is the probability that he will get all vowels? My approach, The number ...
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Is This Conditional Probability Problem asking for Pr(A|B) or Pr(A and B)?

A jar contains black and white marbles. Two marbles are chosen without replacement. The probability of selecting a black marble and then a white marble is 0.34, and the probability of selecting a ...
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Determining probability that blue is part of an outfit based on five shirts and four ties along with a constraint for the possibilities

This problem is from Problem Solving Strategies - Crossing the River with Dogs and Other Mathematical Adventures by Ken Johnson and Ted Herr. I first let capital letters denote shirts, and lowercase ...
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The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
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How to calculate an expected value of maximum random variable function

Let $X \sim U(0,1)$ $Y=\max(X,0.5)$ $Z=\max(X-0.5,0)$ $W=\max(0.5-X,0)$ ask how to calculate $E(Y)$, $E(Z)$, $E(W)$
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Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?
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Random Walk on the 1D Lattice With Barriers

The walk on the 1D Lattice with no barriers can be illustrated where a walker starts at zero (0) on the number line and a fair coin is flipped. If it lands on heads, the walker moves one (1) unit to ...
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How to solve Probability questions? [closed]

I am preparing GRE Test which has Math Part. How to solve probability questions in a real quick ? One marble is randomly selected from a bag that contains only 4 black marbles, 3 red marbles, 5 ...
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Statistics resources with examples for a C.S. student

I'm a computer science student and is fairly familiar with basic probability (calculating the probability of a event occurring, pmfs and pdfs) but I find it very difficult to grasp the concepts of ...
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Stochastic Integral which is almost surely zero at fixed time

This is an exercise from Karatzas and Shreve. Find a $(Y_s)_{s \in [0,1]}$ progressively measurable such that $0 < \int_0^1 Y_s ^2 ds < \infty$ almost surely, and $\int _0^1 Y_s dW_s = 0$ ...
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dice throwing probability. Get face 1 at least one time by throwing a die 6 times.

How can I calculate the following probability: Throwing a die 6 times, what is the probability of having face no. 1 showing at least one time.
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Probability question using binomial coefficient

I am trying to solve the following problem : A biased coin which has P(heads) = p = .7 and p(tails) = q = .3 is tossed 3 times. the coin is tossed in such a way that the outcomes on each toss are ...
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is P(A|B) the same as P(A/B)?

part of a question that I am trying to asks to "use the definition of conditional probability along with results of parts (b) and (c) to calculate P(has disease/positive test)". Now I am confused as ...
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a probability question using percentages

This question is confusing me as I am not used to seeing percentages in a possibility question. ...
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conditional probability of an ace being drawn and then a king

I am trying to study for my statistic exam and not sure how to solve this question. The question is : what is the possibility of the following event: an ace is drawn first and a king is drawn ...
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Fair four sided die is rolled twice, what is the possibility of the sum to be 4 or less?

A fair four sided die is rolled twice, what is the possibility of the sum of the 2 die rolls to be 4 or less? could you please explain in detail
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How to calculate Non-integer raw moment of Beta random variable?

If $B \sim B(1, \beta)$ is a beta random variable. We know that its $k$ raw moment is $$E(B^k) = {\binom {\beta + k} k}^{-1}$$ But how can we calculate $E(B^k)$ when $k > 0$ is not an ...
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How should I generate this random variable?

Suppose we now have a Erlang distribution $b(x;2,1) = x e^{-x}$. According to the definition of Erlang distribution we know that the variance of such a distribution is 2 and we want to reduce the ...
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pdf of the difference of two exponentially distributed random variables

Suppose we have $v$ and $u$, both are independent and exponentially distributed random variables with parameters $\mu$ and $\lambda$, respectively. How can we calculate the pdf of $v-u$?
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Probability and Odds when Gambling!

Here is a 'lunch break' problem from a rather old publication. Devise one set of rules for a dice game, where any number of players and one representative of the bank (mandatory), with one die each,...
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Estimate the sample deviation in one pass

We've learned this algorithm in class but I'm not sure I've fully understood the correctness of this approach. It is known as Welford's algorithm for the sum of squares, as described in: Welford, B.P....
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calculating the expectation

Let $Y$ and $X$ be independent centered normal random variables with variances $\frac{\sigma^2}{1-\theta^2}$ and 1 respectively. How can I compute $$E\left[\frac{YX}{Y^2 + (\theta Y+X)^2}\right]$$ ...
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How to find this probability?

In a company, it's known that $45 \%$ of the employees have a graduate (university), a half are not graduate and make less than $\$1500$per month.Those who are graduate,$\frac{2}{9}$make less than$...
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Let $\lambda > 0$, define the random variable $N$ as follows: where $U_1$, $U_2$, $\ldots$ is a sequence of iid $U(0,1)$ random variables. $U_1 \geq e^{-\lambda}, U_1U_2 \geq e^{-\lambda}, U_1U_2 \... 2answers 78 views How to show that these two random number generating methods are equivalent? Let$U$,$U_1$and$U_2$be independent uniform random numbers between 0 and 1. Can we show that generating random number$X$by$X = \sqrt{U}$and$X = \max(U_1,U_2)$are equivalent? 2answers 126 views Finding a distribution of a random variable generated using a Monte Carlo method I would greatly appreciate if somebody could confirm or negate my result to the following problem. I am especially not sure about "putting it all together" step. Generate$U_1,U_2,U_3 \sim U[0,1]...
We have three events $A$, $B$ and $C$ in question, and given appropriate priors, we derive the posterior $\Pr(A|B)$. Now we want to derive a 'second-order' posterior $\Pr(A|B,C)$ by using the 'first-...