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 of winning a game between players A and B?

The following problem is from A First Course in Probability by Sheldon Ross, and it was assigned as homework by my professor. I was wondering if you guys could help me find a answer to the problem. ...
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2answers
28 views

Factory with $3$ production lines

A factory has 3 production lines A, B and C contributing 20%, 30% and 50%, respectively, to its total output. The percentages of substandard items produced by lines A, B and C are 10, 5, and 2, ...
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Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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1answer
17 views

Why the stopping time makes two parts of Markov chain independent?

Given a finite-length Markov chain $X_0,X_1,...,X_n$ with finite state space, define a random variable $\tau$ as stopping time if event $\{\tau = t\}$ can be determined by $X_0,X_1,...,X_t$ for any $0 ...
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3answers
18 views

multiple rolls probability

I have a $20$ sided die. I roll it $5$ times. Each time I get a number between $1-5$. What are the odds for or against this outcome? (It would be great to have a formula. Intuitively, I don't think ...
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1answer
21 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
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38 views

I couldn't solve this question which is in the book

An orbiting satellite has 3 panels of solar cells, all of which must be activate to provide an adequate power output. The panels function independently of one another. The chance that single panel ...
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3answers
41 views

Upper bound for difference of poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...
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11 views

Two-alternative forced choice

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
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14 views

Probability of $predicate(x)$ if $x \in A \cap B$

I guess my quesion is simple: What is the p($predicate(x))$ if $x \in A \cap B$ When I know that $p(predicate(x))=0.7$ if $x \in A$ and $p(predicate(x))=0.65$ if $x \in B$ If I can't know this ...
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17 views

Poisson process - Number of of arrivals in an 8 hour day with the poisson rate of $\lambda (t)=-(t-4)^2+16 $

The rate of the process is $\lambda (t)=-(t-4)^2+16 $ where $t$ is measured in hours. Find the expected number of arrivals between $t=0$ and $t=8$ I attempted to calculate the integral, where ...
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24 views

Probabilities of Unique Numbers in Roulette

To begin with I know there is a similar question asked and answered, but it is not what exactly i was trying to find (or at least it was only partially answered there) the question that i refer to is: ...
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1answer
21 views

Coin toss problem. $F_{\alpha} = \{\omega: \frac{\#(k\le n: \omega_{\alpha(k)}=H)}{n}\to \frac{1}{2}\}$

$\Omega=\{H,T\}^\mathbb{N}$, so that a typical point $\omega$ of $\Omega$ is a sequence $\omega=(\omega_1,\omega_2,\dots), \omega_n \in \{H,T\}.$ Let $\mathcal{A}$ be the set of all maps $\alpha ...
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3answers
42 views

Expected Value Problem Using Confusing Conditional Probability

I am trying this question: We have a bag with 10 blue jacks and 20 red jacks. We pick 3 jacks from the bag at random and with replacement. We are told that at least one jack is red. Compute the ...
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0answers
21 views

Four of a kind from a 3d pack of cards

I have a $64$-card deck, with $4$ colours - red, green, blue, yellow, $4$ numbers - $1,2,3,4$ and $4$ letters A,B,C,D. So an example card could be yellow2D. I was attempting to calculate the ...
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1answer
23 views

Calculate $P[A,B,C]$ from $P[A,B]$ and $P[B,C]$

I have 3 (not independent) events $A, B, C$ and I know everything about how any two of them correlate. For example, I know: $$ P[A], P[B], P[C], P[A,B], P[A,C], P[B,C], P[A|B], P[A|C], P[B|C], ...
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2answers
22 views

total combinations of divisible sums of $3$

The first $12$ natural numbers are given. Two distinct numbers are selected. What's the probability that their sum is divisible by $3$? This looks very easy. I know answer is $1/3$ but in spite of ...
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21 views

Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts like this $(2,3,5)$ So you buy a ticket and you can win a pot which will multiply your ticket price by the numbers written ahead.The organizer expects a margin ...
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1answer
36 views

Sets and probability

The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum ...
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25 views

What's the difference between the trajectory, the phase portrait and vector field of a matrix?

Take the matrix $$\begin{pmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{3}{4} & \frac{1}{4} \end{pmatrix}$$ as an example. What's the difference between its trajectory(discrete), phase portrait ...
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50 views

Profitable strategy in coin tossing?

There is a coin with a probability $p$ of heads, and $1-p$ of tails. Tosses are independent of each other. When you bet an amount of money $x$, you receive $2x$ if it lands heads, and you lose what ...
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1answer
14 views

Chance of drawing 4 red marbles out of a big bag.

In a bag with an infinite number of marbles, where a third are red, a third are green and a third are blue. Given that you pick $10$ marbles, of which $3$ are blue, what are the chances of picking $4$ ...
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2answers
27 views

What is the probability of observing three or fewer 6s when rolling a fair die twenty times? [on hold]

What is the probability of observing three or fewer 6s when rolling a fair die twenty times? I am trying to figure this out..please be detailed on explaining. Thank you!
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13 views

Conditional probability with sigma-field [on hold]

If $\mathcal{F}_1, \mathcal{F}_2$ and $\mathcal{G}$ are $\sigma$-field, we say $\mathcal{F}_1$ and $ \mathcal{F}_2$ are conditionally independent give $\mathcal{G}$ if $$ P(A\cap B\mid ...
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1answer
25 views

Why total probability is the sum of conditional probabilities?

Consider the following question taken from this link, question number $25$: We have four boxes. Box $1$ contains $2000$ components of which $5$ percent are defective. Box $2$ contains $500$ ...
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1answer
18 views

If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits?

If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits? Intuitively this is not the case, but I am not sure how to formally show this, would ...
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2answers
14 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
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18 views

What is an example where we have convergence in distribution to a constant, but that doesn't imply convergence almost surely?

I have been trying to disprove that if I have a sequence of random variables $X_n$, that $X_n \to a$, where $a$ is a constant, in distribution doesn't imply $X_n \to a$ almost surely. One example I ...
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21 views

Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
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1answer
21 views

Production line probability problem

A manufacturer has six distinct motors in stock, two of which came from a particular supplier. The motors must be divided among two production lines, with three motors going to each line. If ...
2
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1answer
60 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
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2answers
44 views

Find probabilty

I have this table of information: Probabilities: \begin{array}{c|c} .919 & ????\\\hline ???? & .274 \end{array} How do I find the probabilities of the question marks? I thought each row ...
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1answer
31 views

What is the probability that none of these problems are at fault? [on hold]

When a computer goes down, there is a $75\%$ chance that it is due to overload and $15\%$ chance that it is due to a software problem. There is an $85\%$ chance that it is due to an overload or a ...
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2answers
25 views

Is it true in general that $\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r$?

If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that: $$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$? If so, is there a reason why? ...
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15 views

Probability of two people 10 minutes walk away in the same village working together in a different country. [on hold]

I am a teacher of English about to start a new job at a language school in Wroclaw, Poland. My native home is situated in a small village called Kemsing in South East England. Shortly after meeting ...
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1answer
31 views

Two candidates, A & B, are running for president. What is the probability that candidate A beats candidate B?

Candidate A has already garnered 80 votes. Candidate B has already garnered 50 votes. The number of votes a candidate must have in order to win the election is 115. The votes of 5 states are still ...
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3answers
20 views

Joint probability uniform distribution

I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one: We agree ...
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1answer
15 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
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3answers
34 views

Largest possible value of $P(A \cap B)$

Suppose $A$ and $B$ are events with $P(A)+P(B)>1$. Show that the largest possible value of $P(A \cap B)$ is $ \min(P(A), P(B))$. I suspect I'm supposed to use $P(A \cap B) = P(A)+P(B) -P(A ...
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2answers
24 views

Calculating first and second moments for random sums?

Assume that $N$ and $X_1, X_2, \ldots $ are all independent and identically distributed over $(0,1)$ with the density function: $f (x) = cx^2 (1 − x)^2$. An integer–valued random variable, $N$ ...
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31 views

Find $f_Y(y)$ if $f_{X,Y}(x, y)=2e^{−x}e^{−y}$ for $x$ and $y$ defined over the shaded region.

I have no idea how to start this problem. Any help would be greatly appreciated. Thanks
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12 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
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8 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
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20 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
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3answers
65 views

Variance of the random sum of a Poisson?

We have that $N$ and $X_1, X_2, \dots$ are all independent. We also have $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$. Then, we introduce an integer–valued random variable, $N$, ...
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42 views

The probability of the sum of $10$ dice rolls adding up to $57$

So the question is: given that you roll $10$ dice, what is the probability of the sum of the total dice rolls adding up to $57$? I know that there are three ways to do this: Seven die rolls must ...
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2answers
14 views

Find the moments of a binomial conditioned on a binomial?

Suppose that $Y$ has the binomial distribution, $Bin(20, 0.25)$ and conditioned on $Y$, a random variable $X$ that has the binomial distribution, $Bin(Y, 0.5)$. How can I derive the $k$th moment of ...
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18 views

Find the joint and conditional distributions of $Z=X+Y$?

Suppose that $X$ and $Y$ are independent and identically distributed: $$P (X = k) = P (Y = k) = ρ (1 − ρ)^k$$ for $k = 0, 1, \dots$ and let $Z := X + Y$. Find the joint distribution of $(X, Z)$ ...
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3answers
33 views

Meaning of $P(X \in A)$

I have the following problem. I'm struggling a little bit with the expression $P(X \in A)$. My problem is that $A$ is a set, whereas $X$ is a function. I can not really related this two items. Here ...
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26 views

Explicit formula for return probability of simple random walk

Is there an explicit formula for the probability that a simple symmetric random walk on $\mathbb{Z}$ starting from $1$ will not hit $0$ before time $t$?