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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13 views

Expected Value Given Function to Determine Value of Object and Function For Time of Failure

The value, $v$, of an appliance is based on the number of years since purchase, $t$, as follows: $$v(t)={e}^{7-0.2t}$$ If the appliance fails within seven years of purchase, a warranty pays ...
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1answer
78 views

Polynomial Interpolation and Security

Let polynomial $P$ be $P(x)=g(x).(x−β)$, where $g$ is a polynomial and $\beta \leftarrow \mathbb{F}_p$. We evaluate $P$ at some $\textbf{x}=(x_1,..,x_n)$. This gives us $\textbf{y}=(y_1,..,y_n)$. ...
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3answers
39 views

I am looking for some advice on how to calculate the answer to this probability calculation. [closed]

I am looking for some advice on how to calculate the answer to this probability calculation. A box contains $20$ components of which $15$ are good and $5$ are faulty. If $3$ components are chosen ...
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1answer
71 views

How many people are needed in a team to ensure one is at work?

How does one estimate the probability that at least one person from a team is in the office during working hours (9 am - 5 pm)? My assumptions are: There are 3 people in the team There are 253 ...
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1answer
38 views

Proving independence of $A,B$ and $C$ in probability theory

If $A$ is independent of $B$ and $B$ is independent of $C$, then $A$ is independent of $C$. Prove this statement or give a counterexample if it is false. What i tried Form the drawing of the Venn ...
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1answer
43 views

Convergence of random variables taking integer values

Let $X_n$ be random variables taking integer values, and let $X_n\to X$ in distribution. Show $X$ also takes only integer values. $P(X_n=j)\to P(X=j)$ for each integer $j$. $\displaystyle ...
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1answer
44 views

Proving a Conditional Probability Theorem (General Addition Rule)

If $B$ is an event with $P(B) > 0$, prove that the set function $Q(A) = P(A|B)$ satisfies the axioms for a probability measure. , $$P(A ∪ C|B) = P(A|B) + P(C|B) − P(A ∩ C|B)$$ What i tried ...
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1answer
69 views

Probabilities in circular arrangements

For computing probability for a circular arrangement, it should not matter whether we take people in a group as distinct and chairs as numbered, or not, and we should be able to choose as per our ...
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1answer
17 views

Probability of two sets of plates

So about a day or two ago i was driving and saw another car with a license plate almost identical to mine, so i asked myself the following questions: 1)What is the probability that 2 license plates ...
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0answers
32 views

Probabilities in this blackjack variation

Let's say I play blackjack (52 cards, figures count for 10, aces count for 1 or 11) and alone (no dealer). The cards I use for one particular game are always removed at the end of that game and won't ...
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1answer
21 views

Expected value of a discrete random variable

Ok guys, I have a problem with proving this result... I have a random variable $Z$ that can take the values $[1, 2, 3]$ with probability $[\pi_1, \pi_2, \pi_3]$. How can I prove that $\mathbb{E}[Z]=2$ ...
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0answers
47 views

What is the probability that TEAM A will win in both of the following scenarios?

Two teams are in the same league and just finished a season. Assume both teams had played the exact same schedule. TEAM A had a winrate of 90% and TEAM B had a winrate of 70%. If both teams then meet ...
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2answers
41 views

What is the probability that at least one 10 day period contains 6 birthdays if there are 60 birthdays throughout a year?

For all possible contiguous 10 day periods within a single calendar year, what is the probability that at least one of the 10 day periods contains six birthdays if there are 60 birthdays randomly ...
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1answer
40 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
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1answer
35 views

Probability of Dependent Events

Question: what is the probability of any member of a pool of 48 people getting one of 3 prizes if everyone stands an equal chance and no one can win more than one prize. I know how to solve this ...
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1answer
63 views

Equilateral triangle inscribed within nested circles

This seemingly trivial question inspired this one: Somewhere within a circle of unit radius is placed a circle of radius $r, 0 < r < 1$, such that this inner circle's center is uniformly ...
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1answer
50 views

What is the average number of draws (2 cards per draw with shuffles in between) before I had seen all 52 cards in the deck?

On average, how many times would I have to draw two cards from a deck (replacing and shuffling between each draw) before I had seen each of 52 cards in the deck? The process is: Draw the top two ...
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1answer
35 views

How to simplify the equation of combination?

If there are three random variables and three related thresholds, how to simplify the following expression by summation or multiply or other operators? Thank you. ...
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1answer
21 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
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1answer
31 views

Calculating Shannon Entropy for DNA sequence?

I'm following the formula on http://www.shannonentropy.netmark.pl/calculate to calculate the Shannon Entropy of a string of nucleotides [nt]. Since their are 4 nt, ...
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2answers
57 views

Distribution of sum of 2 circular uniform random variables

I hope you can help me resolve the following problem: Let $\Phi_1$ and $\Phi_2$ circular uniform random variables such that $0\leq\Phi_i\leq 2\pi$ (with $i=1,2$). Then the probability density ...
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1answer
15 views

Birthday problem with duplicate days of the month

"Consider the birthday problem except that you ask for duplicate days of the month (assume each month has exactly 30 days)." The answer I got as well as the one given in the book is 7. After 7 ...
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1answer
39 views

Difficult arithmetic trying to follow textbook in probability

Struggling with some steps from my textbook: This is what i have been given: $s(1)=0$ , s(0)=1 and $s(s(x))=x$ (in other words a self-reciprocal function) $$x s\left(\frac{s(y)}{x}\right)=y ...
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2answers
33 views

How does probabilities cope with determinist systems?

Foreword, I have a stronger background in philosophy than mathematics, but I am interested in linking the two topics. So I excuse in advance if this question feel silly; Also please delete/close it if ...
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1answer
32 views

Having the highest value in a interval appear less often

I have an array of size 5. And initially in each index, they are initialized with the value 1. so it looks like this : 1 1 1 1 1 Every iteration, I get a decimal value between 0.0 and 1.0. At the ...
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34 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
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1answer
28 views

# of people that go to a clinic follows a poisson distribution of 4 per day…

I just had an exam and I wanted to discuss a specific question on it. I will do my best to recall the question. Suppose the number of people that go to a clinic follows a poisson distribution of 4 ...
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1answer
37 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
2
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2answers
44 views

expected values of identically distributed random variables

Let $X$ and $Y$ be identically distributed random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then, if I let $F_X$ and $F_Y$ denote the distribution functions of $X$ and ...
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1answer
40 views

covariance matrix of X+Y and X-Y

This question comes up in almost every past paper i do and is worth 10 marks and just can't work it out... Let $X$ and $Y$ have the joint pdf $$f(x,y)= \begin{cases} e^{-y}, \text{if} \ 0 < x ...
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1answer
46 views

Cumulative distribution function implication

How can I prove the following: Let $X$ and $Y$ be two random variables. Suppose that their cumulative distribution functions satisfies $F_X(x)=F_Y(x)$ for all $x$. How can I show that $X$ and $Y$ are ...
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1answer
23 views

Expectation of exponential of integral of absolute value of Brownian motion

Sorry about all the "of"s in the title... here's my problem: I want to compute the expected value of $$ \exp\bigg\{ C \int_0^t |W_s|ds\bigg\} $$ where $W$ is a Brownian motion and $C$ is a positive ...
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2answers
51 views

Which one of the following versions of Bayes' theorem is correct?

I've seen two versions of Bayes' theorem: I've seen this very long version from a frequentist probability class: $$ P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)} $$ where $B^c$ is the event ...
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3answers
56 views

Expected value of die rolls - roll $n$, keep $1$

I know how to calculate expected value for a single roll, and I read several other answers about expected value with rerolls, but how does the calculation change if you can make your reroll before ...
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0answers
19 views

Proving a combinatorics identity (permutations and combinations) [duplicate]

Prove the following identity by interpreting their meaning combinatorially. $$\left( \begin{array}{c} n \\ r \\ \end{array} \right)=\left( \begin{array}{c} n-1 \\ r-1 \\ ...
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1answer
46 views

Am I calculating probability correctly?

Probability was never included in my high school classes, so I'm trying to learn it now from the internet. The downside of this is that you don't get anyone grading your work and catching flaws. This ...
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3answers
44 views

Finding the probability of whether it would rain on weekends

The probability that it will rain on Saturday is 25% and the probability that it will rain on Sunday is also 25%. Is it true that the probability that it will rain on the weekends is 50%. Explain why ...
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0answers
22 views

inequality involving expectation of the maximum

For $X_i \sim$ i.i.d with cdf $F_x$, and $\forall c \in \mathbb R$, then, letting $M_n$ denote the maximum observation $$ M_n \le c+ \sum_i^n (X_i - c) \mathbb I(X_i > c) $$ I proved this by ...
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1answer
46 views

Birthday problem extension question

I have N balls and M boxes. The balls are thrown at random onto the boxes. What is the probability that some box contains at least 3 balls? Based on the Birthday problem, I know how to find the ...
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1answer
51 views

Question regarding Application of Combinations and Permutations (HW Problem)

I have a midterm I am studying for and I don't have the solutions to this homework problem. Can anyone please explain how to do it? I would really appreciate it. Here is the problem: I googled the ...
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0answers
32 views

Joint probability for discrete and dependent variables

Given $\bar{X} \sim N(\bar{\mu}, \sigma)$ is a vector of independent continuous random variables (with identical variance) and $Y_j = ( \bar w_{j} \cdot \bar X + b_j > 0)$ is a set of dependent ...
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1answer
36 views

Given a probability over time, predict when an event will happen

First of all, I'm asking this because I'm writing a game, so this is probably not a typical question in probability. However I'm new to game design so I don't even know what this would be called. ...
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2answers
43 views

Unbiased estimator of $\theta(1-\theta)$:Bernoulli Distribution

Suppose $X_1, X_2, \ldots, X_n$ are a Bernoulli($\theta$) with pmf: $$P(X|\theta)=\theta^X(1-\theta)^{1-X}, \; X \in \{0,1\}$$ Prove or disprove that $\bar{X}(1-\bar{X})$ is an unbiased estimator of ...
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29 views

Almost surely diverging sum implies almost surely diverging sum of conditional expectations?

Suppose $\sum_{n=1}^\infty X_n = \infty$ almost surely for nonnegative $X_n$. Let $\mathcal F_n = \sigma(\{X_0, X_1, \ldots, X_n \})$. Can we show that $\sum_{n=1}^\infty \mathbf{E} (X_n | \mathcal ...
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1answer
21 views

Conditional independence expansion

I have four random variables A,B,C and S. A,B and C are conditionally independent given S. So, I need to obtain P(A,B,C,S) By the chain rule: $$P(A,B,C,S)=P(S)P(A|S)P(B|A,S)P(C|A,B,S)$$ By the ...
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0answers
30 views

The concept of correlation in functional analysis

I am currently reading a book "measure, integral and probability" by Capinski and Kopp. The correlation between random variables $X$ and $Y$ is defined as the cosine of the angle between $X_c$ and ...
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34 views

Probability Question [duplicate]

how would I be able to answer this question? The first box contains 3 white and 7 black balls, and the second box contains 6 white and 3 black balls, A ball is chosen at random from the first box, ...
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3answers
126 views

Why do we subtract [Combinatorics]

I asked Here This question and I am still confused. I got that, for at least one group together there are: $$3 \cdot 9 \cdot \binom{6}{3, 3}$$ But why do we subtract: $3 \cdot 9 \cdot 4$. Lets ...
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35 views

can someone help me to answer this? [Mathematical expectation] [closed]

An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are $\dfrac1{12},\dfrac 1{12},\dfrac 14,\dfrac 14,\dfrac 16$ and $\dfrac16$, ...
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18 views

limiting and monotonic decreasing double sequence of probability measures

I am trying to figure out the behavior of this double sequence of measures. If I have a probability measure $\mu_n$ which is indexed by $n$, and a set of intervals $\mathcal{I}_k$ indexed by $k$ with ...