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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2
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0answers
23 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
159
votes
16answers
28k views

Do men or women have more brothers?

Do men or women have more brothers? I think women have more as no man can be his own brother. But how one can prove it rigorously? I am going to suggest some reasonable background assumptions: ...
7
votes
4answers
515 views

Probablity of being poisoned

You are playing a game in which you have $100$ jellybeans, $10$ of them are poisonous (You eat one, you die). Now you have to pick $10$ at random to eat. Question: What is the probability of ...
5
votes
1answer
1k views

Fingerprint match probability

I am trying to use the formula for the birthday paradox as a reference to figure out an equation that represents the probability of a fingerprint match. Here's the equation for probability of a ...
1
vote
1answer
19 views

Statistical significance and sample size

I have a device that is said to succeed at doing some task at least 99% of attempts, and fails no more than 1% of attempts. If I want to be 98% sure that it achieves that success rate, how many ...
1
vote
0answers
27 views

Bayes' Theorem and Law of total propability for CDF

The calculation of conditional probability is the same for conditional PDF and CDF(according to a number of questionable sources: first, second) (I will use rough notation, with just $x$ and $y$): ...
1
vote
1answer
18 views

Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: ...
42
votes
18answers
74k views

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, ...
1
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0answers
14 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
0
votes
0answers
10 views

Density function of a gaussian vector

Let $X=(X^{(1)}, ..., X^{(n)})$ be a random gaussian vector with mean $m\in\mathbb{R}^{n}$ and covariance matrix $\Sigma$ with $\text{det}(\Sigma)\neq 0$. Prove that the distribution of $X$ is ...
3
votes
1answer
21 views

Total expected time

The first task has a probability of success of $p$, $0 < p < 1$. Assume each attempt are independent. After the first task is completed, the second task (trials also independent) has a ...
1
vote
1answer
30 views

is Cov(X) and Var(X) same? when X is random vector

i'm studying with hogg. introduction to mathematical statistics. and i learned about random vector but i wonder whether Cov(X) and Var(X) is same or not. as intuitive thinking , if X is a random ...
1
vote
2answers
27 views

A stick is broken into two pieces, at a uniformly random chosen break point. Find the CDF.

I'm having trouble understanding how the CDF is found in the solution below: We can assume the units are chosen so that the stick has length $1$. Let $L$ be the length of the longer piece, and let ...
0
votes
0answers
16 views

Ratio of Two Sample Mean of Gamma Random Variables.

Let $X_1,\ldots, X_n$ are iid $\mathrm{Gamma}(\alpha,\beta)$, $Y_1,\ldots, Y_n$ are iid $\mathrm{Gamma}(\alpha,\gamma)$ and independent of $X_i$. What will be the distribution of $\frac{\bar X}{\bar ...
1
vote
1answer
20 views

Finding the marginal distribution for problem with n balls.

I am trying to solve the following problem: A box contains N balls: $N_1\ white, N_2\ black,\ and\ N_3\ red\ (N = N_1 + N_2 + N_3).$ A random sample of n balls is selected from the box (without ...
1
vote
0answers
13 views

Is this an exponential family of distributions? from casella and berger 6.20

I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work? ...
-4
votes
2answers
21 views

probabilities when $n$ independent Bernoulli trials are carried out with probability of success $p$. [on hold]

I don't understand on how to calculate the following probabilities: Let the probability of success be $0.30$. Assume a sample size of $10$ a.The probability of no failures b.The probability of at ...
1
vote
4answers
188 views

Probability of having a Girl

A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl? Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct? ...
2
votes
3answers
49 views

Let $X\sim\mathrm{Exp}(1)$, and $Y\sim\mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. calculate $E(Z)$

Here's a question I'm trying to solve: Let $X\sim\mathrm{Exp}(1)$, and $Y\sim\mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. calculate $E(Z)$ I'm can't understand ...
2
votes
0answers
14 views

Independent coordinates of a gaussian vector

Let $(X^{(1)}, \ldots, X^{(n)})$ be a gaussian random vector. For $i\neq j$, Prove that $X^{(i)}$ and $X^{(j)}$ are independent $\iff \text{Cov}(X^{(i)},X^{(j)})=0$. I'm trying to work with ...
1
vote
2answers
556 views

Probability of the union of $3$ events?

I need some clarification for why the probability of the union of three events is equal to the right side in the following: $$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\cap ...
1
vote
2answers
1k views

An unintuitive probability question

Suppose you meet a stranger in the Street walking with a boy. He tells you that the boy is his son, and that he has another child. Assuming equal probability for boy and girl, and equal probability ...
2
votes
2answers
35 views

An elevator containing five people can stop at any of seven floors.

An elevator containing five people can stop at any of seven floors. What is the probability that no two people get off at the same floor? Assume that the occupants act independently and that all ...
2
votes
1answer
28 views

Finding $\mathbb{P}(A\cup B\cup C)$ under following assumptions

$$\mathbb{P}(A)=0.3$$ $$\mathbb{P}(B)=0.4$$ $$\mathbb{P}(C)=0.5$$ $$\mathbb{P}(B|C)=0.5$$ $A$ and $B$ are mutual exclusive $$(A\cap B=\emptyset )$$ and A and C are ...
2
votes
1answer
24 views

Many urns with colored balls

This is 1.7 from Grimmett's Probability book: There are n urns of which the r th contains r − 1 red balls and n − r magenta balls. You pick an urn at random and remove two balls at random without ...
0
votes
1answer
23 views

Chance of getting a car toy in a chocolate

A mom brings her child every day a chocolate with a toy inside, the toy is random. The boy is happy when he gets a car as a toy. His mom decided to look in which supermarket the probability of getting ...
1
vote
1answer
20 views

Balls in bin. Adding similar coloured ball at every turn.

The bin has 3 black balls and one white ball. In one turn, a ball is picked at random and returned to the bin with another of the same colour. After 60 such turns, what is the probability that the ...
0
votes
2answers
26 views

Finding the distribution of a n tossed fair coin

I am trying to solve the problem: Consider a sequence of n tosses of a fair coin. Let X denote the number of heads, and Y denote the number of isolated heads, that come up. (A head is an ...
5
votes
1answer
2k views

Probability: Drawing Aces from a Deck of Cards

"You are dealt 13 cards randomly from a pack of 52. What is the probability your hand contains exactly 2 aces?" I thought about breaking it down into: ${4 \choose 2}$ = number of ways to choose two ...
0
votes
3answers
52 views

Conditional probability - finding a single event

Context: $P(L) = 0.1$, $P(M | L) = 0.45, P(M | L') = 0.51$ $M =$ Male, $M' =$ Female, $L =$ Left handed, $L' =$ Right handed. 1. Find $P(M)$. So I knew I had $P(M | L)$ and $P(M | L')$ so I ...
1
vote
2answers
31 views

A group of 60 second graders is to be randomly assigned to two classes of 30 each…

A group of 60 second graders is to be randomly assigned to two classes of 30 each. Five of the second graders, Marcelle, Sarah, Michelle, Katy, and Camerin, are close friends. (a) What is the ...
0
votes
0answers
28 views

Question related to expected value I think (Probability) Need help

Hi Im stuck on this question im wondering if someone can help. You dont need to solve it, i think this is an expected value question, but not too sure, which is why im struggling lol. I'd just like ...
3
votes
0answers
12 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures ...
0
votes
1answer
34 views

2 classes in the same classroom each with 100 seats and the same 100 students, find the probability that no one has the same seat for both classes

The question is as follows: Harvard Law School courses often have assigned seating to facilitate the “Socratic method.” Suppose that there are $100$ first year Harvard Law students, and each ...
3
votes
1answer
450 views

How many ways can you choose team of 5 people out of 7 men and 6 women in which there are at least 3 men?

I am confused by this question. I solved it by selecting 3 men first out of 7 men and then selecting 2 people out of 10 remaining person ( 4 men and 6 women ) . So my answer is C(7,3) * C(10,2) = ...
52
votes
10answers
22k views

Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
0
votes
0answers
63 views

Probability problem - no idea where to start [on hold]

It seems like the question I posted earlier was too unclear. So let me rephrase it like this. Suppose there are $n$ players. Each player has a parameter $\theta$ which is uniformly distributed. How ...
1
vote
0answers
44 views

Sum of correlated random variables and the Law of Large Numbers?

Suppose I have a random variable $X$ which can take values on the set $\mathcal{X}=\{1,2,\dots,m\}$ and $X$ is drawn according to the given probability mass function ...
0
votes
0answers
15 views

Derivation of spacing distribution of independent events

A crude approximation of the spacing of energy levels $E_i$ of complex nuclei (like uranium) is that the energy levels appear independently, with known average spacing $D$. I'm trying to understand a ...
3
votes
0answers
23 views

Using the Weak Law of Large Numbers for a product or random variables?

I need to calculate the average of the following quantity: \begin{equation} S_n=\prod_{i=1}^nS(X_i) \tag{1} \label{eq:1} \end{equation} with $S(X_i):=o_{X_i}b_{X_i}$, where each $X_i\in ...
0
votes
1answer
37 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
6
votes
1answer
510 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 ...
2
votes
1answer
35 views

Find the Conditional Expectation $\mathbb{E}[X=k|Y=n]$

Suppose $X$ and $Y$ are stochastic variables with a simultaneous distribution $$\mathbb{P}(X=k, Y=n) = \frac{e^{-1}2^{n-k}}{3^nk!(n-k)!},\ \ \ \ \text{for 0 $\leq$ k $\leq$ n and n $\geq$ 0 }$$ ...
0
votes
0answers
10 views

ACVF: Is Steiner's THM being used here?

I have that {$Z_t$} $\in IID(0,1) $ and if t is even then $X_t=Z_t$ and if t is odd then $X_t=\frac {Z_{t-1}^{2}-1} {\sqrt{2}}$. In the book they write that for even t the ACVF, with $h=0$, is $$ ...
2
votes
2answers
118 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
2answers
29 views

Showing that the multivariate normal density integrates to 1

This is NOT the same as How to show the normal density integrates to 1?. Let $\mathbf{x} \in \mathbb{R}^d$ be a multivariate normal random vector, with $\mathbb{E}[\mathbf{x}] = \boldsymbol\mu$ and ...
1
vote
1answer
713 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
0
votes
1answer
28 views

Indicator function integral

Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space. Let $A, B\in\mathcal A$. Assume that $\mathbb P(A) = 0.5$, $\mathbb P(B) = 0.4$ and $\mathbb P(A\cap B) = 0.1$. Find the integral over ...
0
votes
1answer
405 views

Poisson process - expected number of arrivals in time interval

I came across this question and can't figure out if I'm missing something or if the answer is just "3". Can anyone clarify? Question: "The number of people arriving at a certain "take away" ...
13
votes
3answers
21k views

Expected value of maximum of two random variables from uniform distribution

If I have two variables $X$ and $Y$ which randomly take on values uniformly from the range $[a,b]$ (all values equally probable), what is the expected value for $\max(X,Y)$?