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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33 views

Maximum Likelihood Estimator $P(N=n) = (n+1)(1-p)^np^2$, $n = 1,2,3,\ldots$

An experiment consists of giving a sequence of patients a risky treatment until 2 have died, and then recording $N$, the number who survived. If $p$ is the proportion killed by the treatment then the ...
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0answers
19 views

Proving Independence [duplicate]

The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. Let $n$ be a large integer. A Fibonacci die is a die that has $f_n$ faces. Such a ...
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2answers
79 views

Prove that $P(∅) = 0$

If $P(A\cup B) = P(A) + P(B)$ for every mutually exclusive events $A,B$ then prove *by using only the above property given * that $P(∅)= 0$.
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4answers
267 views

An unexpected application of non-trivial combinatorics

PROBLEM STATEMENT Given two finite sets $A$ and $B$, each containing $s \in \mathbb N$ elements, how many pairs of functions $f \colon A \rightarrow B$ and $g \colon B \rightarrow A$ are there, ...
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1answer
28 views

If you roll a die 5 times, what is the probability that you roll at least one five?

If you roll a die 5 times, what is the probability that you roll at least one five. Is it p(1/5), or do I need to do a bionomcdf, I'm confused.
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1answer
43 views

Convolute two independent general gamma distribution functions

I am trying to create a variable by adding two independent general Gamma probability functions $X$ and $Y$ so that $Z = X + Y$. Both functions have different parameters a and b. $f(x) = b_1 \cdot ...
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0answers
79 views

Poisson distribution-Queueing theory

Vehicles arrive at a junction, in order to swing left, create a line queue ( tail) . The number of vehicle follow Poisson distribution. The length of cycle for the traffic light (for left turns ) is 1 ...
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3answers
44 views

A random variable which distribution is also random

I feel like that question's got an obvious answer, but I somehow missed it during my probability class. There are random variables, which distributions can be expressed if a form of functions - like ...
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0answers
38 views

An outer meassure not being probability measure

I have to prove that an outer measure is not necessary a probability measure. I have this example: Let $\Omega $ be infinity, for every $A \subset P(\Omega)$ $$ \mu^{*}(A) = \left\{ \begin{array}{l ...
1
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3answers
166 views

Fourier transform of random binary vector

Consider a uniformly chosen random binary vector $V$ with $n$ elements. That is we say $V_i = 0$ with probability $1/2$ and $V_i=1$ with probability $1/2$. What is the probability distribution of the ...
0
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1answer
60 views

Probability of getting all four aces playing with 13 people

There are thirteen people sitting on a table. The dealer deals them a card each from his full deck(52 cards). He repeats this until all of his cards have been used. What is the probability that one of ...
1
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1answer
72 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
101 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
44 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 ...
37
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1answer
893 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 ...
1
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1answer
58 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 ...
0
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1answer
49 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 ...
0
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1answer
30 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 ...
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1answer
38 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 ...
2
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2answers
102 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 ...
0
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1answer
37 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
64 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. ...
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0answers
79 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
35 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
43 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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0answers
34 views

Let X be exponentially distributed with mean 1. Let Y|X=x be exponential with mean x. You are interesting in estimating P($XY\leq3$).

I have some difficulties with homework. And I would be glad if you help me. Let X be exponentially distributed with mean 1. Let Y|X=x be exponential with mean x. You are interesting in estimating ...
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0answers
33 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. ...
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1answer
35 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
votes
1answer
76 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
50 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. ...
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2answers
38 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
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1answer
70 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
48 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
34 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
66 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 ...
1
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1answer
34 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
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1answer
26 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
56 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
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1answer
44 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. ...
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3answers
232 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
28 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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0answers
25 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
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0answers
40 views

Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
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3answers
52 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 ...
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3answers
137 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, ...
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1answer
60 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
88 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 ...
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1answer
56 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
36 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 ...
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1answer
187 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 ...