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
14 views

Probability of choosing same number

There are four friends – Adam, Bella, Christopher and Drew. All of them are asked to choose any number in their mind. Now what is the probability that every one of them has the same number in mind? ...
0
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1answer
9 views

Quick question concerning the sum of random number of random variables given mean and variance and average

$\DeclareMathOperator{\cov}{cov}$The problem is: Let $X_1, \ldots, X_n$ be independent random variables with mean $µ$ and variance $σ^2$. Let $X¯$ be the average of these n random variables. Find the ...
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0answers
22 views

Maximum value of the product of probabilities

I came across a confusing probability problem. It reads as follows: Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(\cdot)$ denotes the ...
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0answers
24 views

Question about package of cookies that is a random variable

The weight in grams of package of cookies is a random variable with expected value of $300$ grams $\color{blue}{A)}$ assume that X is normally distributed with standard deviation of $15 $ grams ...
2
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0answers
42 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
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0answers
39 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
0
votes
1answer
47 views

Probability as a function of time

I was really wondering when I have to select any one out of the n options available - the probability of selecting A (let's say) is 1/n. But then I'm confused. When I (or anyone/anything else) bring ...
0
votes
1answer
19 views

Probability set function of the random variable $X$

Let a point be selected from the sample space $S = (0,10)$. Let $C \subset S$ and let the probability set function be: $$P(C) = \int_C \frac1{10}\ \mathsf dx$$ Define the random variable $X$ by: ...
0
votes
1answer
29 views

Probability of 4-number matching in a lottery in two different situations?

In some lottery, 7 numbers are drawn and each of them from numbers ${\{1, \dots, 45}\}$. To win "Division 6" means to have 4 of 7 drawn numbers. The order of drawn numbers doesn't matter. My ...
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2answers
33 views

Reverse Bernoulli Trial?

I'm struggling to figure out how to do what I think would be called a reverse Bernoulli trial, essentially: How many coin flips must I make to have a 75% change of getting three heads? First of ...
0
votes
1answer
21 views

Probability of an event if the sample space has identical elements

Suppose we have a box, with only one small hole. Suppose 10 distinct black balls and 20 distinct white balls are put in the box. Now, in a random draw of 1 ball, the probability that the ball drawn is ...
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0answers
7 views

How to illustrate that the first-fit algorithm for bin packing problem uses STRICTLY MORE bins when only one object becomes larger?

In the bin packing problem, objects of different volumes(lie in $[0, 1]$) must be packed into a number of bins(each of capacity 1). The first-fit algorithm attempts to place the item in the first bin ...
2
votes
3answers
59 views

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?
-1
votes
1answer
22 views

Risk Reduction equation

If, after a point in time, your risk of an event falls by 50% in 1 year and then by 100% in 15 years, can someone help me with the equation that will look at your risk reduction to date for any given ...
0
votes
0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
1
vote
1answer
38 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
-1
votes
1answer
51 views

A conditional probability question [on hold]

Let A and B two events and if $P(A)=0.5$ and $P(B)=0.4$ what is the $P(B\mid A^C)$?
3
votes
1answer
25 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
4
votes
2answers
67 views

“Mastermind”-esque safe opening problem.

I read this interview question for a trading job and it seems quite difficult. What is the technique to solving it? You have a safe with six digits and a light. You can input a code, if you have ...
2
votes
2answers
28 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
0
votes
1answer
15 views

Inner product estimator - random variable

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure. If I generate randomly $N$ realizations of $x_i$ following ...
2
votes
1answer
25 views

Coding Theory - Probability that word received has distance of at most 1?

Suppose the codeword x = 101101 is transmitted over the binary symmetric channel, with symbol error probability p. What is the probability that the word received has distance at most 1 from x? ...
3
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0answers
30 views

Convolution of probabilities

It is a well known fact that for a random variable $Z=Y_1+Y_2+...+Y_n$ where $Y_i$ are independently distributed then the probability density function of $Z$ is the convolution of the density ...
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0answers
20 views

Probability of word appearing in a new document given the probability it appears in earlier documents [on hold]

Suppose I have a set of keywords and 10 documents . I can count the frequency and probability of a given keyword occuring in each of these documents . What will be the probability of the same keyword ...
3
votes
1answer
35 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
votes
1answer
21 views

Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
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0answers
27 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
2
votes
0answers
20 views

Total variation distance and couplings

The total variation distance between two measures $\mu$ and $\nu$ can be shown to equal the infimum over all couplings $(X,Y)$ where $X\sim\mu, Y\sim\nu$ of $P(X\neq Y).$ What is the supremum of ...
0
votes
0answers
23 views

How many uniform samples are needed to hit every element

Let $D$ be the uniform distribution over $\{1,\ldots,n\}$. How many draws from $D$ (asymptotically) are needed such that with high probability (say $2/3$) all $n$ elements were drawn at least once?
1
vote
1answer
42 views

Circuit probability question regarding sum of a random number of independent random variables

Suppose we have n circuits that operate in a home. Each one will live according to an exponential random variable with rate λ. If X denotes the time at which a circuit first dies (i.e. the first circuit ...
0
votes
2answers
32 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
0
votes
1answer
28 views

Find probability of a Poisson Process

I have a Poisson process $N(t)$ with $\tau$ for customer arrival in a shop. $N(t)$ is spllitted with two types of arrival (male and female). It can be shown that the process is a combination of two ...
3
votes
0answers
53 views

Why is $F$ continuous?

Why is the function: $F: P(\mathbb R) \to \mathbb R$, $F(X) = \int_X e^{-x} dx$ a continuous function? How to prove such a thing? Does it even make sense to talk about the continuity of such a ...
-1
votes
2answers
24 views

Conditional day distribution probability

Let $X$ be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so $X$ takes values 1, 2,..., 7, with equal probabilities). Let $Y$ be the next day after $X$ (again represented as ...
2
votes
1answer
34 views

Random variable to the power of minus one?

I have a definition, it goes as follows: $\Pr$ is probability. $X$ is a random variable. $x\in\mathbb{R}$ $$Pr(X = x) = \Pr(\{ \omega\in\Omega \mid X(\omega)=x\})$$ So for example for a dice of 6 ...
1
vote
1answer
52 views

Coin and Lottery Question

Attempt: For the first part I presume I use bayes' theorem? For the second part, I can't count the number of ways of such a sequence. Thanks.
2
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0answers
18 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
2
votes
2answers
53 views

Calculating probability for forming a triangle

I am having trouble coming up with a solution for this problem: There is a stick of unit length. We break it into two parts. Now, we pick the bigger one and break it into two parts. I want to ...
2
votes
3answers
40 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
0
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0answers
14 views

Simulate ICA Source Signal

I am using the fastICA package in R for a matrix of time series information. However, if I wanted to simulate the process for risk management purposes how exactly could I do this? For example lets ...
0
votes
2answers
27 views

The random variable $ Z = 1-F(X)$

I will formulate the theorem (with no proof) if $X \in \mathbb{R}$ is a random variable with continuous distribution function $F$ then the random variable $Z = 1-F(X)$ has a uniform distribution on ...
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votes
2answers
30 views

Does the order in a circular arrangement matter?

I posted a question a while ago: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. My question here is: imagine a ...
3
votes
1answer
61 views

Play until one is bust

If player A and B have $a$ and $b$ millions pounds respectively, where $a,b$ are natural numbers. They play a series of games in which the winner receives one million pounds from the loser (draws ...
1
vote
1answer
32 views

Custom Weighted Formula

I'm in need of a mathematical formula that will be ultimately utilized in any programming language that would give me a value that I could ultimately sort or rank by. I have 2 variables: variable 1 = ...
3
votes
4answers
58 views

Is conditional probability always meaningful

Problem: A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement. Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the ...
2
votes
1answer
119 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
6
votes
2answers
240 views

$\{X_n\}$ are iid random variables with symmetric distribution

Let $X_1,X_2,\ldots,X_n$ be iid random variables with symmetric distribution. Show that $$P\left(|X_1+X_2+\cdots+X_n|\ge \max_{1\le i\le n}|X_i|\right)\ge \frac12.$$ I was trying it for $n=2$. ...
1
vote
1answer
34 views

Definition of n independent event and example

Given a finite set of events $A_1,\dots,A_n$, the events are said to be independent if and only if for any subset of indices $I$ we have: $$\mathrm{P}\left(\bigcap_{i\in I} A_i\right)=\prod_{i\in I} ...
9
votes
3answers
1k views

Kelly criterion with more than two outcomes

I want to calculate the Kelly bet for an event with more than two possible outcomes. Suppose the following game: A jar contains $10$ jelly beans. There are $7$ black jelly beans, $2$ blue jelly ...
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vote
4answers
6k views

Chance of getting six in three dice

I am having a hard time wrapping my head around this and am sure that my answers are wrong. There are three dice. A. Chance of getting exactly one six on the three dice. (1/6) * 3 = 1/3 B. Chance ...