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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20 views

Compute maximum R-squared of time series

Suppose I have a timeseries $(x_i)_{i\in[0,N]}$ and create another timeseries as follows, $$ y_i = x_i +\epsilon_i, \qquad \epsilon_i \sim \mathcal{N}[0,n\sigma_X] $$ Where $\sigma_X$ is the ...
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2answers
27 views

How do you describe the sample space of a lottery (49 balls, 6 draws without replacement)?

Sample space = {1,2,3...49}^6 where sample point w_i is removed after each draw doesn't sound that mathematical. Is there a correct way to write it?
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1answer
51 views

Sufficiency of $X_{(n)}$ for random sample of scale uniform variables.

Consider a random sample $X_{1}, \dots, X_{n}$ where $X \sim \mathrm{unif}[0, \theta]$ for $\theta \in (0, \infty)$. Usually we prove that $T = X_{(n)}$ is a sufficient statistic for $\theta$ by ...
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2answers
53 views

Probability of circle inside square

We pick a uniformly chosen random point on a unit square (a square with unit side length) and draw a circle of radius 2/22 around the point. Find the probability that the circle lies entirely inside ...
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3answers
76 views

Typos is a book. Probability

On the first 400 pages of a book, you notice that there are, on average, 10 typos per page. What is the probability that there will be at least 3 typos on page 401? Think about what assumptions you ...
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1answer
38 views

Demonstrate uniform continuous distribution using tangible items?

What is the best way to explain "equally likely" in continuous uniform distribution to an audience using tangible or everyday items?
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1answer
41 views

Finding density and distribution functions [closed]

I have been trying to understand probability by attempting past paper question and I have been stuck on this question all day and night. I am not quite sure how to go about finding the functions ...
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2answers
22 views

Why does |A1|=18 and |A2|=6 in this rolling a die random variables question?

Rolling a die. Let $X_i$ be the score on the $i$-th roll of the die, then $X_1, X_2, \dots$ form an independent sequence of random variables. When rolling the die twice, we have sample space $\Omega ...
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1answer
35 views

Notation for a statistic, or function of a random variable

A statistic is a function of random variables, so it is also a random variable. Suppose we have a collection $X = (X_1, X_2, \dots, X_n)$, where $X:\Omega \to \mathcal{X}^n$. There are two common ...
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2answers
34 views

Why is $P(A\mid B)=\sum_{i=1}^n P(AB_i\mid B)$?

I am trying to show that, if $B_1,\ldots, B_n$ is a partition of $B$, then $$P(A\mid B)=P(A\mid B_1)P(B_1\mid B)+\cdots+P(A\mid B_n)P(B_n\mid B).$$ A hint given for solving this problem is that ...
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1answer
45 views

A problem on probability concerning distributions of particles

I saw this problem in An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). I am quite intrigued by the problem. Here it is: Suppose that a particle is equally likely ...
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1answer
59 views

What is the probability of two people telling the truth?

I have a similar problem described in this question: How to find the probability of truth? The question reads: A and B are independent witness in a case. The probablity that A speaks the truth is ...
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1answer
22 views

Variant on Law of Iterated Conditioning

By the law of Iterated Conditioning, we know it is true that for sigma-algebras G and H, with $H\subseteq G$ it follows that $E[E[XY|G]|H] = E[XY|H]$ for random variables X, Y on $(\Omega, H, G)$. ...
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0answers
30 views

probability problem, communication system of n components.

A communication system consists of $n$ components. Each of those components operates independently from others with a probability $p$. The total system operates only when at least half of these ...
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1answer
40 views

Joint PDF given, find missing constant

I have a relatively simple question regarding a joint PDF given to me. There are 2 random variables X and Y, with the following joint PDF. a) Find the value of a. I have attempted to set up the ...
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0answers
32 views

Log of characteristic function

I am looking at a proof related to CLT and I have difficulties understanding a condition in the theorem. Basically, the theorem is trying to show that $\frac{d^m}{d u}ln[\varphi(u)]\leq k \beta_n $ ...
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1answer
30 views

Calculating the number of arrangements of items into compartments?

I'm dealing with intro probability theory here, and am a bit perplexed by the logic around arranging items in boxes. When arranging a small set of objects in a series of boxes, I am not seeing why the ...
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1answer
72 views

Expected Value of a Continuous Random Variable (Example from Sheldon Ross's Book)

This is an example in the book (A First Course in Probability by Sheldon Ross). A stick of length 1 is split at a point U that is uniformly distributed over $(0,1)$. Determine the expected length of ...
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1answer
20 views

Finding the variance of X with a variable p between 0 and 1

Here is the full question (and note this is a question on my homework, but I just don't understand a portion of it): Suppose a random variable $X$ has the support of $S_X = \left\{0,1\right\}$ with ...
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1answer
129 views

Probability that random walkers meet

I was wondering about a question about Random Walks. I came across various papers where the probability of 2 random walkers in 1 dimension and 2 dimension starting at the same point and returning to ...
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1answer
58 views

Number of possible passwords with $6$ to $8$ characters

Question: Passwords on a computer are $6$ to $8$ characters long, where each character is either an uppercase letter or a digit. How many possible passwords are there if each password must contain at ...
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3answers
445 views

infinite monkey problem - probability of an infinite sequence containing an infinite sequence [duplicate]

Note: This question is specifically about when the infinite monkey theorem is extended to reproducing an infinite sequence (as oppose to a finite one) I was browsing wikipedia, and came across the ...
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2answers
42 views

Concept of random sample? I have a truly problem understanding it.

I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). Suppose that we have a box with ...
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1answer
43 views

Binomial? probability of two producing defective articles machines.

Suppose that machine A produces (on a daily basis) twice the articles that machine B produces. However, 4/100 of the articles produced by machine A are defective while 2/100 of the articles produced ...
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0answers
58 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
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1answer
17 views

Computing the mean of a Gaussian random variable

I am trying to compute the mean of a Gaussian random variable with $X(μ,σ^2)$ I am stuck on this integral as it seems I cannot use integration by parts nor substitution. I searched on other ...
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1answer
45 views

next nine births at a hospital all being girls determine whether the event is independent or dependent and find the probability

next nine births at a hospital all being girls determine whether the event is independent or dependent and find the probability I found the event as independent and probability being 1/9 is this ...
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1answer
34 views

The solution to this joint distribution problem is too terse for me to understand.

I was wondering if I could get clarification on the following problem: We know that $\sum_x\sum_y f(x, y) = 1$. Then $4\theta_1 + 6\theta_2 = 1$. I understand that $P[X = 1] = ... = P[Y = 4] = ...
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2answers
90 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
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2answers
41 views

When do we need combination factor?

Say I want to draw 4 balls from a big ball pool, with 3 kinds of colors: red 50%, white 30%, black 20%. Now, I draw 1 ball of each time for 4 times, each time with replacement(or the pool is big ...
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1answer
39 views

How to rewrite this probability formula?

In probability, I know that $$P(C=1|W=1)=$$ $$\dfrac{P(C=1,W=1)}{P(W=1)}$$ But what if I have variations like: $$P(C=1|W=1,R=1) and P(C=1,R=1|W=1) and P(C=1 or R=1|W=1) and P(C=1|W=1 or R=1)$$ ...
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2answers
96 views

In the card game “Projective Set”: Compute the probability that $n$ cards contain a set

In the game of Projective Set, it turns out that any seven cards contain a projective set. For fewer than 7 cards, how can we determine the probability that one or more sets exist (in terms of the ...
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1answer
19 views

What is the difference/similarirty between P() and P[] in the discussion of Probability Theory?

Do Parentheses and Square Brackets mean the same things in the discussion of Probability Theory? For example, are P(X=0) and P[X=0] mean same things? Do they have other uses?
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1answer
71 views

The name for predicting future rolls of dice based on the past

My friends and I were playing a game where you roll dice and you bet money on what picture it's going to land on and I began reasoning with myself that if I tallied up what pictures the dice landed on ...
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2answers
49 views

Use of permutations and combinations for a conditional seating arrangement

I've got a problem where there are 6 seats in a row, with 6 people to be seated. I need to figure out all possible seating arrangements, given that two people are married and are to be seated next to ...
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1answer
40 views

Constructing a piecewise probability density function

Here I'm trying to construct a probability density function in the form $$f(t) = \begin{cases} at, & t \in [0, 5) \\ b\sqrt{t}, & t \in [5, 20]\text{.} \end{cases}$$ Of course, ...
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1answer
88 views

Expected Value of the Maximum Number of Heads in n Flips

How would one go about finding the expected value of the maximum number of consecutive heads when flipping a coin $n$ times? For small $n$, it seems easy to brute-force it (i.e. when $n = 3$, the ...
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2answers
35 views

Probability of n random numbers under a threshold

If we choose N random numbers between 0 and 1 (uniformly distributed), what is the probability that at least n will be under some threshold t? For example I want to know if I have a list of 1,000,000 ...
2
votes
1answer
32 views

Highschool Probability: Chances of player winning

I tried finding the probability of getting a W which turned out to be 1/2 and then multiplied it by the probability of getting a number greater than 1. However my answer turned out to be 0.3, if you ...
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1answer
3k views

How many people would you need in a room to ensure with 100% probaility that three have the same birthday?

I am vaguely aware of the Pigeonhole principle and I understand that you would need 367 people to ensure that two people have the same birthday. I think that it may be required to have 734 people in a ...
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1answer
25 views

finding variance of gaussian distribution from mean

The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute ...
2
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1answer
41 views

Stoppage time for sequence of uniform random numbers with a recursively shrinking domain

Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which ...
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1answer
33 views

What is the probability that a player will receive a red card over the course of the season

I am trying to calculate the probability that a goalkeeper will get at least one red card over the course of the season. There are on average 2.75 red cards (in total for all goalkeepers) per season. ...
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1answer
24 views

Find the distribution with the following Laplace transform.

Is anybody aware of the distribution whose Laplace transform is the following. \begin{equation} \mathbb{E}[e^{-tX}] = \frac{e^{-t}}{(1+2t)} \end{equation} Note: The Laplace transform of the ...
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3answers
150 views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than ...
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2answers
89 views

4 girls and 8 boys are randomly divided into 3 groups of equal size

4 girls and 8 boys are randomly divided into 3 groups of equal size What is the probability that there will be at least one girl in each group?
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0answers
74 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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1answer
43 views

Is c parameter or constant (random variable X with given density)

problem: is c constant or parameter solution for this is to $ \int_{1}^{2} cx^2 dx = \frac{7c}{3} $ $ \int_{2}^{3} cx dx = \frac{5c}{2} $ Until now I understand what is going on; next (I am ...
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2answers
34 views

Find probability that $X\sim Geom(p)$ is even

I have to find the probability that $X$, which has geometric distribution with success parameter $p$, is even. For all $n\in \mathbb{N_1}$ $$P(X=2n)=(1-p)^{2n-1} p=(1-p)(1-p)^{2n-2} p=(1-p) ...
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0answers
32 views

Does bitwise-XORing substrings results in a uniform distribution?

Let's say I have an integer $k$ whose bit string representation can be exactly divided into $l$ substrings of length $\log_2(m)$. Let's call each one of these substrings $B_i(k)$, for ...