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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2answers
23 views

Two dimensional Gaussian adding one dimensional Gaussian

Based on this question, from one dimensional Gaussian to two dimensional Gaussian, I have the following question. Any help is appreciated. Suppose $x$ is one-dimensional Gaussian distributed, with ...
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1answer
44 views

How can I determine the expected cost to win a game that has a 1/100 chance of winning?

If a game has a 1/100 chance of winning, and the cost to play is $1.00. What is the expected cost to win the game? Initially, I thought the answer was ${$1.00 \over 1/100}$. But this isn't really ...
9
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1answer
453 views

length of Gaussian Random Vector

Suppose I have a random vector $x=[x_1,...,x_k]$ s.t. $x∼N(\mu,\sum)$. How is the length or magnitude of $x$ distributed? I know that if $k=2$ and $\sigma_1=\sigma_2$ and $\sigma_{12}=0$ ($x_1$ and ...
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1answer
94 views

Help needed to derive combinatorics formula.

I am having troubles understanding a combinatorics formula. I would appreciate any ideas or hints, leading to an explanation how this formula might be derived. I came across the formula reading a book ...
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1answer
47 views

Probability of drawing 6 specific letters from a set of 144 with 21 draws

I know similar questions have been asked before, but I am struggling to find a method to calculate the probability for the situation below: I have a set of 144 scrabble tiles with the following ...
2
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1answer
119 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
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1answer
20 views

Probability of one node given all the others in a bayes network

For a bayes network which has $n$ nodes, $X_1, X_2, ... , X_n$. Is there any efficient way to calculate $P(X_i|X_1,X_2,...,X_{i-1},X_{i+1},...X_n)$, without constructing the full joint distribution?
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1answer
37 views

Binomial Distribution; Using a mean to suggest probability of success.

and thanks for taking the time to look at my question. This was for a homework task, on binomial distribution, which i thought i understood. A consignment of china mugs is packed for ...
3
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2answers
237 views

Expected value of number of tosses until we get $k$ tails and $k$ heads

Let $k$ be a fixed positive number. We toss a normal coin until we get at least $k$ heads and at least $k$ tails (not necessarily consecutively). Let $X$ be number of needed tosses. Find distribution ...
1
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1answer
61 views

from one dimensional Gaussian to two dimensional Gaussian

Given $x$ is a random variable, which is one-dimensional Gaussian distributed, such that $$ x \sim \mathcal N(\mu, \sigma^2).$$ Given $\theta$ is some constant, what would be the density function for ...
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0answers
13 views

Solution verification on standard normal random variable.

The problem I am solving is as follows: A certain IQ examination has a score that is normally distributed with mean $100$ and standard deviation $14.2$. What is the probability that a randomly chosen ...
2
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2answers
305 views

Probability of selecting distinct numbers in ascending order

A relatively simple problem is stumping me on this :( Given a set of 4 distinct numbers that can be chosen from [0-9], what is the probability that after selecting, the first two numbers would be in ...
1
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1answer
38 views

How to convert expectation to integration

$S: \{1,-1\}^n \rightarrow \{0,1\}$ and $E(S(x))=p$, Where $E$ denotes the expectation, and is taken over $x$ , where $x$ is uniformly distributed on $\{-1,1\}^n$. Then how to prove the following, ...
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1answer
50 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
1
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1answer
118 views

Extracting balls from a basket until all have the same colour

Consider a basket containing $n$ red and $m$ blue balls. At each step choose a (random) ball in the basket and put it out. Keep doing this until all the (remaining) balls have the same color. Let $X$ ...
1
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2answers
63 views

Is it reasonable to calculate $P(X_1 = X_2)$ given that $X_1$ and $X_2$ are continuous random variables?

To be specific, suppose $X_1$ and $X_2$ are independent exponential random variables with parameters $\lambda_1$ and $\lambda_2$; what is $P(X_1 = X_2)$? According to section 5.2.3 of the book ...
1
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1answer
108 views

How do you calculate a binomial distribution with k > R as opposed to k = R

I'm given the formula: $\displaystyle P(X = k; n, p) = \binom {n}{k} * p^k * q^{n-k}$ And we need to work out the binomial coefficient by hand, instead of using C(n,r). So I have a question: "Some ...
3
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2answers
54 views

Probability question related to coin tosses

In an exam I gave recently, the following question was asked: A fair coin is tossed $10$ times and the outcomes are listed. let $H_i$ be the event that the $i^{th}$ outcome is a head and $A_m$ be the ...
40
votes
1answer
1k views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
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2answers
245 views

Edge percolation on $\mathbb{Z}^2$: probability that two neighbouring vertices are connected?

I'm considering edge percolation on $\mathbb{Z}^2$ with parameter $p$, so that edges are present with probability $p$. Is it known how to express the probability $P(p)$ that $(0,0)$ is in the same ...
0
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1answer
31 views

Ordered Partitions

I'm having trouble figuring out exactly how one would get the answer for a question such as: "how many ways can 9 concert tickets be divided between 4 concertgoers, such that one person receives 3 ...
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1answer
31 views

A question about probability (1)

Given a circle $C$ of radius $a$, a point is selected at random within the circle, what is the probability that the point is the centre of the circle?
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3answers
320 views

Probabilities When we use the Combinations and when not? nCk vs nPk

Question 1: I am working on probabilities and on some exercises the solutions either use the nCk or just n. I want to find a method to understand when I have to use the nCk and when not to. Example ...
2
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1answer
28 views

Broken trucks at a road

If three trucks break at locals random distributed of a road with lenght $L$, find the probability that $2$ of those trucks are not at a greater distance than $d$, fot $d \leq \frac{L}{2}$ My ...
0
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2answers
56 views

Binomial and Normal Distribution Problem - Check solution

Whooping cough is a highly contagious bacterial infection...About 80% of unvaccinated children who are exposed to whooping cough will develop the infection, as opposed to only about 5% of vaccinated ...
2
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1answer
41 views

probability with replacement

A box contains 4 red, 3 black and 2 white cubes. A cube is randomly drawn and has its color noted. The cube is then replaced, together with 2 more of the same color. A second cube is then drawn. ...
0
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1answer
40 views

Truth Tables. (If Q, then P)

Suppose people from Tracy always tell the truth and people from Livermore always lies, You meet Diego and he says,' If I came to San Jose on the train, then I am from Livermore ."is it possible ...
0
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3answers
561 views

How to put probability density function in C++?

I have a random variable X that has a probability density function of f(x)=x^(-1/2)/2 for all x between 0 and 1. double RANDOM; I need to give a value to RANDOM, that accepts the PDF All that I ...
3
votes
1answer
57 views

Finding the number of possible ways to 'paint' a geometry

Problem: Find the number of possible ways to paint $n$ sectors of a disk with $n$ color brushes to differentiate all $n$ sectors. (i.e., you cannot paint an adjoining sector with the same color) ...
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1answer
27 views

Probability of zero events in large population given known event rate

I have a group of 1444 patients that have been included in a group of different trials. The trials found that none (zero) of the patients had a specific complication (independent). The known incidence ...
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1answer
31 views

Fair coin tossed $3$ times. $X$ is # of H's on $1st$ and $2nd$ tosses, $Y$ is # of H's on $2nd$ and $3rd$ tosses. Write Joint PD.

Suppose a fair coin is tossed $3$ times. Let the random variable $X$ represent the number of heads on the first and second tosses and let $Y$ represent the number of heads on the second and third ...
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0answers
38 views

Interesting Probability Game With Uneven Return Scenarions

Let say we play a game. The game which spans the course of 5 trials. The game is as follows. You either receive 100 points or 40 points as a final payout. The only time a payout of a 100 points occurs ...
1
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1answer
28 views

Distribution of $U + Y$ $\mod 1$

Let $U$ ~ Unif$[0,1]$ be uniformly distributed on $[0,1]$. Let $Y$ be some random variable, independent from $U$. What is the distribution of the random variable defined as $X=U+Y \mod 1$? Can ...
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0answers
44 views

Random permutation with a scientific calculator

I have 8 people whom I want to divide into 2 groups. The allocation must be uniformly at random, i.e., every person must have equal probability of joining either group. We came across a situation ...
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3answers
64 views

probability? Marbles question

A bag contains two blue marbles and two red ones; two marbles are drawn at random. (a) What is wrong with the following argument? Because there are four possibilities – (red, red), (blue, blue), ...
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0answers
41 views

If $A=\{R_1\le\frac{1}{2}\}$ and $B=\{R_2\le\frac{1}{2}\}$, find $P(A\cup B)$

Let $(R_1, R_2)$ have the following density function $f_{12}(x,y)=\cases{ 4xy & \text{if } 0\le x,y\le1, x\ge y\cr 6x^2 & \text{if } 0\le x,y\le1, y>x }$ If ...
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1answer
233 views

Continuous probability about meeting between two friends [duplicate]

My friend and I are hoping to meet for lunch. We will each arrive at our favorite restaurant at a random time between noon and 1 p.m., stay for 15 minutes, then leave. What is the probability that we ...
0
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1answer
30 views

Dealing with this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. $f_1>0$ is a positive constant and $f_2,f_3,f_4\ldots$ are positive functions of one variable. ...
0
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1answer
62 views

Log-space probability of a log-space probability not occurring

Normally the probability of some probability $p$ not occurring would be $1-p$. However, I'm working with very small probabilities and therefore must work with $p$ in $\log$ space (Ie. I'm working with ...
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1answer
31 views

Computing expectation by conditioning

A coin is tossed as long as it shows heads. Let $X$ be the number of heads before the first tail. After that you throw a dice $X$ times. Let $Y$ be the number of sixes. calculate $E[Y]$ I've ...
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3answers
203 views

Expected value of number of draws

We have $5$ number in a bag: $(1,3,5,7,9)$. We draw one from the bag and then put it back. We do this until the sum of the numbers can be divided by $3$. Whats the expected value of the number of ...
1
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1answer
191 views

CDF of a continuous random variable with discontinuity

With a piecewise PDF defined as $$f(x)=\begin{cases} 1/2 & -3 < x < -2 \\ e^{-2x} & x>0\end{cases}$$ Would the CDF of this function between $-2$ and $0$ be $0.5$, even ...
0
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1answer
262 views

Poisson distribution, (conditional) probability question

Suppose Bob receives on average one call per night from his father. Find the probability that 7th January was the third night this year (starting on 1st January) when no night calls were received. I ...
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0answers
24 views

Expected number of errors in a magazine page

The expected number of typographic errors in a page of a certain magazine is 2.What is the probability that an article of 10 pages has 2 errors? My attempt: I thought that if $X$ is the r.v that ...
1
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1answer
54 views

Probability with same number of heads and tails

I have this question that I don't know where to begin, any help will be greatly appreciated. Consider the following random experiment. Toss a coin until the same number of heads and tails have been ...
1
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1answer
58 views

Solving a general integral (expectation of some variant of exponential distribution)

Suppose $X$ is distributed exponentially with parameter $\lambda$. Its pdf is $\lambda e^{-\lambda x}$, and the calculation of its expectation is straight forward: $\mathbb{E}(X) = \int_0^\infty ...
0
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1answer
30 views

Question regarding Poisson random variable

I'm reviewing a question I did for the Poisson random variable and I can't remember how I got the the answer to part b). The problem goes as follows: "Suppose that the number of typographical erros on ...
0
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3answers
37 views

Probability of having less than 3 females if…

Assuming that half the population is female and assuming that 100 samples of 10 individuals are taken, how many samples would you expect to have 3 or less females? Can someone please ...
2
votes
1answer
71 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
1
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1answer
19 views

Normal Distribution how $N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2) $

I read an expression Could someone explain the step $N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2) $ ?