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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-6
votes
2answers
41 views

Chances of this…

9 people sat in a circle. They wrote their name on a piece of paper, folded it over and placed it in a hat. The hat was shuffled to mix up the pieces of paper. The first person picked out the name ...
0
votes
2answers
50 views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
2
votes
1answer
21 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
1
vote
1answer
32 views

“Inverse” of nondecreasing, right-continuous function?

Suppose $F : \mathbb{R} \to \mathbb{R}$ is a nondecreasing and right-continuous function. Define $G : [\inf F,\sup F] \to \overline{\mathbb{R}}$ by $G(p)=\inf \{ x : F(x) \geq p \}$, with the ...
1
vote
0answers
15 views

Consider a random walk where $p \neq 1/2$, where the starting point is random and has a binom distn. Find the probability of absorption at $N$.

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
votes
3answers
29 views

Probability of a Rare Event Occurring within a Certain Number of Times

I'd like to know how to find the probability of an event occurring, given the probability of that event, within a certain number of chances for it to occur. For example, say that once every year ...
2
votes
1answer
26 views

probability question that just seems to easy to be the case

the game of mastermind starts in the following way: one player selects four pegs, each having six possible colors, places them in a line. the second player then tries to guess the sequence of colors. ...
-1
votes
0answers
15 views

Convergence in law and probability

I have a succession of random variables $\{X_n\}$ with $P(X_n=3)=1/n^2$ and $P(X_n=4)=1-1/n^2$. It's defined $Y_n=nX_n$ and i have to prove the convergence in law and in probability. I found ...
2
votes
2answers
42 views

What is the intuition of why convergence in distribution does not imply convergence in probability

For me its very counter intuitive (that convergence in Probability and Distribution are not the same), because, conceptually, if two random variables have the same distribution, then they should be ...
0
votes
0answers
33 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
0
votes
1answer
25 views

Standard deviation: calculating how polarizing a question is

I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer. Answer a: $30\%$ Answer b: ...
4
votes
1answer
316 views

Probability that a given Poisson variable samples greater than its mean $\lambda$, provided $\lambda > D$

Given a random variable $X \sim \text{Poisson}(\lambda)$ such that $\lambda > D$, with $\lambda, D \in \mathbb{N}$, what is the probability that a sample obtained from $X$ is greater than ...
4
votes
2answers
46 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
2
votes
1answer
526 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
4
votes
4answers
102 views

Casino turns 50% of your losses into “free play”, are odds in your favor?

As a limited-time promotion, if you gamble during your first week at this casino, and you suffer a net loss of money, the casino will give you half of your losses (up to a certain amount) as "free ...
1
vote
1answer
31 views

Independence of Random Variables

If $X$ and $Y$ are independent random variables so are also the random variables $f(X)$ and $g(Y)$ for $f$ and $g$ measurable and bounded functions. The independence of $X$ and $Y$ implies: ...
1
vote
1answer
26 views

Multiple examination of a result (probability)

A performs a task and submits the result to B and C for examination. B confirms the result. C thinks the result is wrong. The reliability of A is 0.7, for B is 0.8 and that of C is 0.9. (reliablity = ...
0
votes
2answers
52 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
0
votes
1answer
26 views

What is the probability that on a given day, the number of half gallon containers provided is enough?

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. ...
6
votes
2answers
59 views

what is the probability that the circumcircle of 3 point

Mary picks any three non-collinear points inside a given circle, what is the probability that the circumcircle of these 3 points will be covered by the original circle? This is from a test ...
0
votes
0answers
15 views

Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional $\phi$ on the Wasserstein ...
-6
votes
0answers
24 views

probability of getting lucky in exam? [on hold]

In an examination, you are given a choice to pick up a chit, which has a question, there are ten of those chits(randomly arranged), only half you have prepared(you know all the question but you're ...
2
votes
1answer
447 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
0
votes
1answer
40 views

Check My Work on a Poisson Process/Distribution Question

I'm just curious if my work is correct, and if not, where I made a mistake. My Task: Cars arrive according to a Poisson process with a rate of 12 per hour. (1) What is the probability that the ...
-1
votes
0answers
50 views

Expectation of an interval

Given $g(\theta) := Pr\{X\leq\theta\leq Y\}$ with $Y\geq X$, what is $E[Z]$ where $Z:= Y-X$ ? Also $X{\not\perp}Y$ Progress: $$X\leq\theta\leq Y\Rightarrow \{Z \geq \theta-X\}\cap \{Z\geq\ ...
0
votes
1answer
46 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
1
vote
0answers
14 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
0
votes
0answers
19 views

Combination of historical and current data in statistics

I have a general question about a statistical matter. Lets assume there exists a true and unique probability $p$ such that an event $X$ happens in the next 12 months. There is some information about ...
4
votes
1answer
34 views

Coin Flips and Hypothesis Tests

Here's a problem I thought of that I don't know how to approach: You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
-4
votes
0answers
36 views

Four points inside a rectangle [on hold]

We randomly choose 4 points inside a rectangle.What is the probability that they lie in the same half ?
2
votes
3answers
32 views

2 restaurants located randomly

any help on following question will be much appreciated. Q. Suppose that $2$ restaurants are going to be located at a street that is $10$ km long. The location of each restaurant is chosen randomly. ...
4
votes
1answer
51 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't play nicely with ...
0
votes
1answer
13 views

Exercise on iid sequence of uniformly distributed random variables (and LLN).

I'm trying to solve following problem: Let $X_{1}, Y_{1}, X_{2}, Y_{2},\ldots$ - iid, from uniform distribution on $[0,1]$, $f\colon[0,1]\rightarrow[0,1]$ be measurable and $Z_{j} = ...
0
votes
0answers
21 views

Probability of a train journey

A trip from south east London to Southampton consists of three journeys: bus journey to Crystal Palace station, train journey from Crystal Palace to Clapham Junction, train journey from Clapham ...
1
vote
0answers
18 views

Easy Question from Application: Estimate for transition probabilities of random walk - finding a coupling

SHORT VERSION: Find appropriate Coupling Suppose we have a random walk on the natural numbers, where we go to the left with probability $p_L \geq \frac{1}{6}$, to the right with probability $p_R\leq ...
2
votes
3answers
25 views

Horse racing question probability

Been thinking about this for a while. Horse Campaign length: 10 starts Horse Runs this campaign: 5 Horse will is guaranteed to win 1 in 10 this campaign Question: what is the Probability of ...
2
votes
2answers
39 views

9 room probability and expected value

I got the following question: In a house with 9 rooms. There is 1 mouse that is looking for some food. This can be found in 2 rooms, but there are also 2 cats, these are in different rooms. When the ...
0
votes
1answer
14 views

Question about assigning probabilities to elementary events

Let $(\Omega,S)$ be a sample space with a probability function $P$. Then, the book by Rohartgi that I am reading says that: if $\Omega$ is uncountable, one cannot assign positive probability to ...
2
votes
2answers
32 views

Binomial distribution central moment calculation

If for a binomial distribution the mean is $4$ and variance is $3$, find th $3^{\text{rd}}$ central moment. I understand that the first and second central moments are mean and variance ...
1
vote
1answer
511 views

Probabilities Pick 3 Lottery

In a Pick 3 lottery three numbers are drawn from three separate sets of numbers, between 0 and 9. Matching 2 winning numbers, one of them duplicated, can be done in 3 ways, according to all lotteries ...
0
votes
0answers
21 views

Challenging probability and statistics problems?

The books I have on the subject lack entertaining problems. Can someone suggest a book with more challenging problems - perhaps not at olympiad level, but a bit easier. Thanks very much!
1
vote
3answers
75 views

Problem with the Birthday Problem

I recently learned of the Birthday Problem in probability theory, which essentially states that it only takes 23 people in a room to have a 50% chance that 2 of those people have the same birthday. ...
1
vote
0answers
25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
1
vote
1answer
25 views

Probability of dominating set in random balanced tournament

I'm trying to estimate some probability in a random tournament, and I know that what I have is false, as it leads to contradicting results published some 40 years ago. But I don't know where the ...
1
vote
1answer
30 views

gaming - How to calculate odds of roulette “Strategy”

I know that this strategy shouldn't work, but I can't seem to get the math to work to make it fail, and when I model it; it succeeds. I'm obviously missing something, but can't see what. In American ...
1
vote
1answer
21 views

Poisson process problem

Problem statement below... Customers of a store frequent it at a rate of 0.75 customers per minute following a Poisson process. Exactly one customer goes into the store during a 2 minute span. ...
0
votes
1answer
364 views

Greatest of three random variables

Assume that we have $3$ not equal random variables $(A, B, C)$. If we know that $$Pr(A>B)=x, \quad Pr(A>C)=y, \quad Pr(B>C)=z$$ What is $Pr(A$ is the greatest one)? I know that $Pr(A$ is ...
1
vote
1answer
71 views

conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
3
votes
5answers
23k views

What is the difference between independent and mutually exclusive event? Explain with example and counter example.

Two events are mutually exclusive if they can't both happen Independent events are events where finding out about one doesn't change the probability of the other. Is it fine ? what should be its ...
-1
votes
1answer
19 views

How to show that the 3 events are independent but not pairwise independent [on hold]

Say the experiment is tossing 2 dice and the events are A - first die is a even number B - sum of both dice is 4 C - outcome of the two dice differ by at most by 2 What I got is $(A \cap B ...