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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Very basic probability problem

Firstly, apologies if this question is too trivial or just plain inappropriate for this site... I want to catch a ball. I have exactly two chances to catch the ball. The ball is thrown in such a way ...
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1answer
25 views

Probability of picking from a sublattice

Short version The set of partitions of a four-element set forms a lattice. Suppose that I pick $n$ times from the set of tri- and bipartitions (i.e., the top element = quadripartition and the ...
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0answers
44 views

Coupon Collector's problem with possibility of failure

In the standard problem, there are n items and you need to collect all of them. I'm curious about what would be the expected number of times you need to draw if instead you also have a chance to draw ...
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3answers
48 views

How to find $E[\sqrt{X}]$ given only a distribution function?

I'm given a continuous distribution function $F(x)$ and I should assume that $X$ is some random variable. My goal is to find the expected value of $\sqrt{X}$. I'm very bad at probability but this is ...
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1answer
122 views

Do we need to use continuity correction if we use CLT to do normal approximation

In a hotel, large number of cups and saucers are washed everyday. The number of cups that are broken each day while washing averages $2.1$. The number of saucers broken each day averages $1.6$, ...
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1answer
22 views

Conditional probability new variable

Variables $X_1, ... , X_N, N$ are independent $X_i - \exp(1)$ and $N$ have geometric distribution with parameter $\frac{1}{2} $ . We have a new variable $Z= \min(X_1, X_2 , \ldots , X_N)$ I must ...
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1answer
38 views

does uncorrelation imply mean independence?

Suppose we have $ cov(\epsilon_i , \epsilon_j)=0$, $ E(\epsilon_i)=E(\epsilon_j)=0$ and they have the same finite variance. Can we deduce from these assumptions that $E(\epsilon_i |\epsilon_j)=0$ ? ...
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2answers
180 views

Combinatorics of a tournament where one wins by taking either three games in a row or four in total

Two teams play each other repeatedly until either one of them wins three games in a row or one of them wins a total of four games. What are all the ways in which the tournament can be played? What ...
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1answer
39 views

Explicit CDF associated to Gamma PDF [closed]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...
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1answer
32 views

Cryptography probability

62% of plaintext messages have even parity. 56% of odd plaintext messages have ciphertext with even parity. 48% of even plaintext messages have ciphertext with even parity. What is the probability ...
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1answer
15 views

Given an estimator, how do you find the large-sample confidence interval?

I'm given three possible estimators for the mean of a random sample of size $n$ from a population with unknown distribution, but it is known that the mean is $\mu$ and the variance is $\sigma^2$. I ...
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2answers
25 views

Conditional and Joint Probability

Box 1 : 5 white and 2 black balls Box 2 : 2 white and 1 black balls Box 3 : 2 white and 3 black balls One box is selected at random and one ball is drawn from it. What is the probability that it will ...
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1answer
21 views

How to prove (or disprove) a non-negative r.v. with finite expectation also have finite second moment?

The problem is formulated as follows. If $X$ is a non-negative random variable and $EX<\infty.$ Is it true that $EX^2<\infty?$ It seems that squaring wouldn't make much difference because $X$ ...
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2answers
16 views

Probability of tagged individuals

If there are $20$ tagged cows in a field of $200$, and I take $10$ cows (no replacement), what is the probability of exactly $2$ cows being tagged? Is this a binomial distribution, where $P = 0.1$ ...
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0answers
14 views

Entropy of non-ergodic process

Two coins have been kept in a box, One is fair while the other is biased. One coin is picked. The probability of either coin being picked is equal. The picked coin is then tossed again and again to ...
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1answer
40 views

Example where $E \subset S$, $E \neq \emptyset$, and $P(E) = 0$?

From what I understand, this situation isn't possible since any non-empty subset of S will have a non-zero probability. What am I missing here?
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2answers
42 views

Probability that A meets B in a specific time frame

A and B are going to the math palace. A arrives between 12:00 and 13:00 o'clock and stays for 10 minutes, whereas B arrives between 12:00 and 14:00 o'clock and leaves immediately. The time point in ...
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1answer
29 views

Find Monte Carlo Variance When Expected Value is not Known

I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same. I ...
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2answers
88 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
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1answer
28 views

Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
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3answers
105 views

Probability of impossible event.

There is question in my book: Probability of impossible event is? After reading the question my instant answer was $0$ and that was the answer given. But then i thought other way, question is ...
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1answer
33 views

probability of universally true events

I've got an argument with few friends about a probability question.Is there any difference between below 2 statements? & what are their respective answers? 1) What is the probability that 1 is a ...
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1answer
40 views

Estimating how much two probability distributions differ

I have two probability distributions A and B. First I would like to estimate how much they differ. In this regard I use as metric the Jensen–Shannon distance (i.e. the square root of Jensen–Shannon ...
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1answer
27 views

A group of X people of different heights line up in 2 rows.

A group of X people of different heights line up in 2 rows. One row is behind the other one. There are an equal number of people in both rows (X is even). The only rule is that any person in the back ...
1
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1answer
21 views

Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
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1answer
94 views

Gambler's ruin and coin toss

Edit 3. Fixed question to be more clear and include current solution Problem Two players player 1 and player 2 plays a game of fair coin flipping. Player 1 starts with $A$ coins and Player 2 with ...
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1answer
23 views

Notation: Building a set from sequences of random variables, some a.s. equal

For $1 \leq i \leq n$ let $(\psi_{ij})_{1 \leq j \leq n_i}$ be sequences of random variables. Is there a better notation than $$\{\psi_{ij} : 1 \leq i \leq n, 1 \leq j \leq n_i\}$$ to build a set ...
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3answers
232 views

Probabilities related to the sum of four dice [closed]

Suppose we have 4 fair six-sided dice of different colours and faces numbered 1,2,...,6 are rolled independently. (a) How many ways can a total of i. 4 ii. 5 iii. 6 be obtained? (b) Compute (to ...
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1answer
33 views

Probability and independence (joint PMF)

Given a joint PMF, if I wanted to see if X and (Y - X) are independent, do I check: $F_{x,(y-x)} = F_x{(x)} * F_y{(y-x)}$? This is just my understanding given what I've found on wikipedia: ...
0
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1answer
34 views

Probability of Fair Die

Hi everyone I have a question about probability: Fair die thrown two times, final score is calculated as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and ...
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0answers
29 views

Calculate distance between 2 probability distributions

I have 2 probability distributions composed by 12 elements ...
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1answer
32 views

Poisson distribution excercise, finding $\lambda$

On a motorway, in a 5 minute interval it is equally likely whether a car exceeds the speed limit or not. What is the probability that exactly 3 car exceed the speed limit in this 5 minute? I need to ...
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1answer
33 views

Question about the Central Limit Theorem

The version of the CLT in my book states that if $X_1,...,X_n$ is a random sample, with mean $\mu$ and standard deviation $\sigma^2$, then $W=\frac{\bar X-\mu}{\sigma/\sqrt n}$~$N(0,1)$ as ...
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1answer
35 views

Show that this Markov chain is recurrent

So I have a Markov chain on the nonnegative integers such that, starting from $x$, the chain goes to $x+1$ with probability $p$, $0<p<1$, and goes to state $0$ with probability $1-p$. I'm ...
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2answers
47 views

Find $E[N]$, where $N = \min\{n>0: X_n = X_0\}$

Let $X_i$, $i\geq 0$ be independent and identically distributed random variables with probability mass function $$ p(j) = P\{X_i=j\},\; j=1,...,m,\;\sum^{m}_{j=1}P(j)=1 $$ Find $E[N]$, where ...
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1answer
69 views

Product of two distribution functions.

Let F and G be two distribution functions, does the product FG still a distribution function?
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1answer
79 views

Probability: Distinguishable vs Indistinguishable

So there are 5 red balls and 4 blue balls in an urn. We select two at random by putting them in a line and selecting the two leftmost. What's the probability both are different colors if the balls are ...
0
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1answer
45 views

How do you determine the likelihood of all 0s or all 1s in an infinite coin toss?

A {0,1} faced coin is tossed an infinite number of times, the probability of 1 in any toss being equal to p $\in (0,1)$. The resulting sequence $\omega$ is parsed into consecutive blocks of ...
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1answer
166 views

Combinatorial Analysis: Fermat's Combinatorial Identity

I was looking through practice questions and need some guidance/assistance in Fermat's combinatorial identity. I read through this on the stack exchange, but the question was modified in the latest ...
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1answer
40 views

Detailed explanation of the derivation of the expected number of coin tosses

In this post, I cannot understand the logic behind the approach nor can I devise an approach myself. Can anyone be so kind so as to give me a detailed explanation as to what is being done? More ...
0
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1answer
32 views

How to do you compute the probability a record occurring in a sequence of independent experiments?

Consider a sequence of independent experiments, each of which produces a random integer in N with the probability mass function ${p_k}$. The pmf is the same for all the experiments and also strictly ...
0
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1answer
36 views

A bag contains 5 red marbles and 6 white marbles. Two marbles are drawn in succession without replacement.

The first marble drawn is red and the second is white Both marbles drawn are red.
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1answer
18 views

How to determine the probability of even number of ones in a fair coin toss?

A {0,1} coin is tossed m-1 times (assume m is >= 2). We have a set of events $A_1,A_2,A_3,...$ where $A_k$ is the event that the kth outcome is 1 (where k=1,2,...,m-1). $A_m$ is the event that ...
0
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1answer
24 views

Angle of three independently chosen points on a cricle

If a,b,c are three points on a circle (viewed in $\mathbb{R}^2$ not disc) chosen independently and uniformly,and p(x) is the probability that at least one of the angles of the triangle formed by the ...
0
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1answer
34 views

Probability of 7 Card Hand with All Different Ranks?

Suppose we're dealt a 7 card hand from a standard 52 card deck. I'm trying to find the probability that all 7 cards are different ranks (that is, no two cards share the same rank). I know the ...
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1answer
38 views

Conditional probability with balls in urns involving discards

I found this problem in a statistics book, and I'm wondering if my solution is correct. "You and a friend play a game involving 20 balls in an urn, of which 1 is red and 19 are white. The game is ...
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1answer
182 views

Probability: student passing an exam by randomly guessing (no calculator)

Assuming you can't use a calculator, how do you estimate the answer to the following problem? Suppose an exam has 40 questions, all multiple choice. Each question has 5 choices and you need 20 ...
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1answer
25 views

Expected value of Cumulative Hazard

Define $T=\min(T^0,C)$ where $T^0$ is the failure time and $C$ is the censoring time. Define the failure indicator $$\delta = \begin{cases} 1 & \text{if $T^0\leq C$}\\ 0 & \text{if $T^0> ...
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0answers
21 views

Can an a.s. (almost surely) finite random variable be a.s. UNbounded?

I thought that if a random variable, $\eta^2$, is assumed to be a.s. finite, then $\eta^2$ must be a.s. bounded. In the Martingale Central Limit Theorem in Hall & Heyde, they assume this: "let ...
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1answer
38 views

Weighted Coin Toss Probablity

Suppose two weighted coins are tossed. The first is weighted so that it comes up heads with probability $\frac{1}{3}$. The second is weighted so that it comes up heads with probability $\frac{1}{4}$. ...