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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1answer
401 views

There is 10 people that pick-up random number between 1 to 20

There is 10 people that pick-up random number between $1$ to $20$. More then one person can pick up same number (i.e. the pick-ups are independent). What is the probability that the minimum number of ...
0
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2answers
192 views

Sum of two normal distributions, $Z=X+Y$

Having trouble with this probability question: IF $X\sim\mathcal{N}(1,1)$ and $Y\sim\mathcal{N}(1,2)$ are two normally distributed random variables with means and standard deviations as indicated, ...
2
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1answer
61 views

Confirming the correctness of Combinatoric permutation formula

for the ordering and distinguishable non-empty, is it $$ (n)k = n(n-1)(n-2)\cdots(n-k+1) $$ for no ordering distinguishable non-empty, i got $$ \sum_{i=0}^{n-1}(-1)^i\dbinom{n}{i}(n-i)^k $$ please ...
46
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1answer
1k views

How likely is it not to be anyone's best friend?

A teenage acquaintance of mine lamented: Every one of my friends is better friends with somebody else. Thanks to my knowledge of mathematics I could inform her that she's not alone and ...
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1answer
164 views

Gambler's ruin, probability of loss, infinite turns

A gambler starts with $\$1$ and bets $\$1$ every turn of a game, where he has the probability $p$ to obtain $\$2$ and $1-p$ to obtain nothing. If $p<1/2$, what is the probability he will eventually ...
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1answer
69 views

Does reducing 512-bit blocks to 128-bit hashes lead to 1/4 chance of collision?

This is a quote from a cryptography book called Implementing SSL / TLS Using Cryptography and PKI By Joshua Davies. MD5 operates on 512-bit(64 byte) blocks of input. Each block is reduced to a ...
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1answer
46 views

Binomials for getting probability of standard deviation

I have the following problem which I am stuck on the second part. Suppose that $30\%$ of all students who have to buy a text for a particular course want a new copy whereas the other $70\%$ want a ...
2
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1answer
117 views

Central limit theorem

I'm stuck on this idea from my lecture notes: Using the Central Limit Theorem, $$Y^{(n)}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{Y_i-\mu}{\sigma}$$ Then $$\lim_{n\rightarrow\infty}P(Y^{(n)}\leq ...
4
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2answers
1k views

Poisson Distribution when only given using mean

I'm doing the following homework problem and am unsure of whether or not my answers are correct. This is my first time working with Poisson distribution and I want to make sure I am doing it ...
1
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1answer
382 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
1
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1answer
118 views

NCAA bracket and binomial coefficients

Given that March Madness is almost here I was trying to figure out the probability of constructing a perfect bracket if you just flipped a coin for every game. I came up with two possible solutions. ...
6
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1answer
283 views

Spectral gap of mixture of Markov chains

Context Let $P$ be the transition matrix of an irreducible, aperiodic, discrete-time Markov chain. The spectral gap is given by $$\xi = 1 - \lambda_\max$$ where $\lambda_\max = \max\{\lambda_2, ...
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1answer
60 views

Average # of tracks that must be heard on a CD, whose player's shuffle function can play the same song >1×, before all tracks of CD heard. [duplicate]

Suppose you have a CD of $n$ tracks. Your CD player's shuffle function is broken; it selects a random song, possibly even the one(s) already played, before all tracks are played. How many tracks ...
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1answer
46 views

Probability problem with example

Show that if an a sample space all results dont not have zero Probability, then the following mathematical property is is true: $P(A\cup B) = P(A) + P(B) \implies AB= Ø$ Then find an example that ...
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1answer
55 views

Symmetrisation of function

Consider the probability space $\Omega = \{-1, 0, 1\}$ with the $\sigma$-algebra of all possible events and a probability measure $P$. Consider also the smaller $\sigma$-algebra $$F = \{\emptyset, ...
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0answers
52 views

The mean of $\mu_{P}(\theta)=\frac{1}{Z}P(x|\theta)$

Consider a parametrized probability measure $P(x|\theta)$, that is for each $\theta\in[a,b]$ it is a valid probability measure on $x$. Denote $f(\theta)$ its mean and $\Sigma(\theta)$ its variance. ...
0
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1answer
44 views

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$?

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$? Standardizing, $P(\frac{X - 5}{\sigma} < \frac{-1 - 5}{\sigma}) = 0.1587$ $P(Z < ...
0
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1answer
81 views

On strong convergence of Sum of square of spacings

I need the proof of strong convergene of $$(n+1)\sum_{i=1}^n W_i^2$$ as $n ‎\rightarrow‎ \infty,$ where $W_1, W_2, \dots W_n$ are spacings from uniform distribution. Thank you anybody in advance
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2answers
93 views

Proof : An event is independent from every other event iff its probability is 0 or 1

As said in the title I need to prove that an event is independent from all other events iff its 0 or 1. One side is pretty simple, if I assume the event is 0 or 1 probability the answer is immediate. ...
3
votes
1answer
401 views

Conditional expectation of conditional expectation

I have a question about conditional expectation. I have always problem with that... It is a step of a proof that I just don't get... I appreciate any help! I have the random variable $$B=S+ ...
0
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2answers
64 views

Number built from {1,2,3,4,5} given that all digits must appear, what is the probability that the digit appears twice will appear one after the other

Build number of digits - 1,2,3,4,5 given that all digits must appear, what is the probability that the digit appears twice will appear one after the other? what I tried to do is at first find $\Omega$ ...
3
votes
1answer
76 views

$A$ and $B$ are events, is $A|B$ an event?

$A$ and $B$ are events, is $A|B$ an event? Can I write an event in this form $A|B$ ? $Pr(A|B)$ means given $B$ happens, what is the probability of $A$ happen. Since $Pr(\cdot)$ is a measurement, so ...
2
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0answers
85 views

Flipping two coins; $X$ is how many times first coin is flipped until heads, $Y$ is how many times second coin is flipped until heads

The first part of the question is to find the probability that the two coins take the same number of flips to land on heads, which I found to be $1/3$.The next part, which I'm stuck on, it to find the ...
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1answer
52 views

Probability as it relates to flipping a coin

My friend and I flip a coin when we have lunch to see who buys. My friend did not win a coin flip the whole month. I came up with this plan to give him a better than 50/50 chance of winning. That is ...
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2answers
94 views

Probability of a specific outcome of a “Lottery” machine

I play a particular phone game in which you battle against monsters with monsters of your own. One of the methods to get these monsters is an in-game lottery machine, that at the pull of a lever and ...
2
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0answers
40 views

Expectation of a Uniform PDF

How do I find the expectation of the following pdf? $f(x,y) = 1/\pi r^2$ , where $x^2+y^2 \leq r^2$ I've tried to integrate it on the bounds $-\sqrt{1-x^2}$ and $\sqrt{1-x^2}$ for $\int ...
0
votes
1answer
62 views

Balls in an urn

Suppose that there are 6 balls in an urn, 2 red and 4 white. There are two players. The first player draws a ball at random. If the ball is white then it is replaced and the other player draws, and so ...
2
votes
1answer
145 views

Mary can answer 20/25 problems correct… simple probability

Question: A teacher gave his class $25$ problems and told his students that he would select $10$ of them to put on their midterm. Mary can figure out how to answer $20$ of the problems, what is the ...
0
votes
1answer
205 views

String probability (with conditional prob and combinations)

I'm having trouble with the questions below, all relating to string probability. I'll write the problem and then provide my work for my (incorrect) answer. Please help me figure out what I did wrong. ...
3
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3answers
1k views

the probability that a patient recovers from a rare blood disease

The probability that a patient recovers from a rare blood disease is 0.4.. if 15 people are known to have contracted this disease what is the probability that (a) at least 10 survive (b) from 3 to ...
2
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0answers
66 views

Supremum of empirical process

Suppose $\hat{F}_{n}$ is the empirical distribution function based on a sample $(X_{1},\ldots,X_{n})$, where each $X_{i}$ has distribution function $F$. Also, suppose that the distribution of ...
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3answers
99 views

Finding the probability

Suppose I throw 3 balls and each ball is equally likely to land in one of 4 buckets. What's the probability no bucket has more than 1 ball in it? I know the answer is 3/8 but for some reason I can't ...
3
votes
3answers
500 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
2
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1answer
406 views

Draw two cards what is the probability the second is higher than the first? Is my approach correct?

I've seen similar questions posted here before but I was wondering if my method/answer was correct My reasoning was let's say you draw a 2 as your first. Card there are 12 cards with higher values, ...
2
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1answer
326 views

Is this a poorly worded probability question? Unsolvable?

The question says: "For a recent year, 0.99 of the incarcerated population is adults and 0.07 is female. If an incarcerated person is picked at random, find the probability that the person is female ...
0
votes
1answer
223 views

Probability of drawing balls

A box contains 12 balls of which 4 are white and 8 are red. Three players A, B, and C draw a ball in succession replacing each ball after it is drawn. The first player to draw a white ball wins the ...
0
votes
1answer
89 views

Expectation of a product of (many) 1-dimensional Brownian motions.

Let $0=t_0<t_1<t_2<\ldots$ be a sequence of positive reals. Denote by $B(t)$ the 1-dimensional Brownian motion with time $t$. It is easy to show the the expectation of the product of two ...
5
votes
1answer
222 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
3
votes
1answer
78 views

Expected Time for n Independent Prisoners to Escape

Suppose there are $n$ prisoners, and each day every prisoner independently has a probability $p$ of escaping. What is the expected length of time before all prisoners have escaped? Someone asked ...
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1answer
118 views

Probability distribution “similar” to Gaussian.

Does there exist a distribution A other than Gaussian such that: 1) linear combination of random variable from A is distribution A 2) easy to integrate, for example find entropy Thank you
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2answers
51 views

Are X and |X| independent? where $f(x)=\exp(-|x|)/2$

The density function of $X$ is $f(x)=\frac{e^{-|x|}}2, x\in(-\infty, +\infty)$. Are $X$ and $|X|$ independent? My thought is: Let $Y=|X|$, so $f(y)=e^{-y},y\in(0,+\infty)$ Then try $P(X\le x,Y\le ...
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2answers
348 views

Finding cdf of difference of two random variables from a joint distribution

I have that: $$f_{X,Y}(x,y) = \begin{cases} 8xy, & \text{if $0<y<x<1$} \\ 0, & \text{otherwise} \\ \end{cases} $$ I'm trying to solve for the cdf of a random variable $W$, where ...
0
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1answer
62 views

Two players $A,B$ throw two dice…

Two players $A,B$ throw two dice. A throw first, and they throw it in turns (i.e. $A,B,A,B,A...$). If $A$ gets sum of $10$ at the dice he wins, if $B$ gets $9$ - he wins. What is the probability ...
4
votes
2answers
235 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
2
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2answers
58 views

Generate random sample with three-state Markov chain

I have a Markov chain with the transition matrix $$\pmatrix{0 & 0.7 & 0.3 \\ 0.8 & 0 & 0.2 \\ 0.6 & 0.4 & 0}$$ and I would like to generate a random sequence between the three ...
2
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0answers
136 views

Multiple Hypergeometric Distributions

I need to figure out a problem which involves multiple hypergeometric distributions. Referring to the Urn problem, the problem can be described like the following: We have $n$ urns $u_1,…,u_n$. Urn ...
1
vote
2answers
101 views

Are there hidden events?

Consider the sample space S = {a, b, c, d} and a probability function Pr : S ->R on S. De fine the events A = {a}, B = {a, b}, C = {a, b, c}, and D = {b, d}. You are given that Pr(A) = 1/10, Pr(B) = ...
1
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1answer
139 views

When is a minimum distance decoder also a maximum likelihood decoder?

It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming ...
0
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1answer
260 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
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1answer
48 views

Computing $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$

Let X be a uniformly distributed over $[0,2]$, and $Y$ to take values from $[0,\infty]$, how do we compute $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$. My attempt: $$ E\left[X\,\middle|\, X ...