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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
9 views

Prove that for a sequence of people sets $S_1,…,S_d$, $\Delta_i \not = 0$ for all people

We have $k$ people $p_1,...,p_k$, and $d$ people sets $S_1,...,S_d$, where the sizes of $S_1,...,S_d$ can vary between $1$ and $k$ (so each $S_1,...,S_d$ is a set of some people from ...
0
votes
0answers
4 views

Maximum load is $O(\log\log n/\log\log\log n)$

There are $n$ bins labeled $0,1,\ldots,n-1$, and $\log_2n$ players. Each player chooses a starting location $k$ uniformly at random, and places one ball in each of the bins $$k\bmod n,k+1\bmod ...
1
vote
1answer
5 views

Poisson approximation to bound probability of balls in different bins

Suppose $n$ balls are thrown randomly and independently into $n$ bins. What is an upper bound that all balls land in different bins using Poisson approximation? The exact probability is $n!/n^n$, ...
0
votes
0answers
6 views

Game of Red balls two drawings are made, which rule would you choose if playing the game, rule A or rule B?

In the game of redball two drawings are made without replacements from a bowl that has four white ping pong balls and two red ping pong balls. The amount won is determined by how many ping-pong balls. ...
1
vote
1answer
10 views

Last two bins have same number of balls

If we throw $n$ balls independently and randomly into $n$ bins, what is the probability that the last two bins have an equal number of balls? We can write that as the sum of the probability that each ...
5
votes
2answers
593 views

Regression towards the mean v/s the Gambler's fallacy

Suppose you toss a (fair) coin 9 times, and get heads on all of them. Wouldn't the probability of getting a tails increase from 50/50 due to regression towards the mean? I know that that shouldn't ...
1
vote
1answer
15 views

Application of Law of Iterated Expectations

Could you please explain something from a text I am reading? We're given that $E(\epsilon_i)=0$ and $E(\epsilon_i x_i)=0$ for $i\geq 1$. Write $g_i=\epsilon_i x_i$ and we further know that $$ ...
0
votes
1answer
8 views

Velocity Obstacles — Probabilistic Collision Cone concept

I have been working with the Velocity Obstacles concept. Recently, I came across a probabilistic extension of this and couldn't understand the inner workings. Source: Recursive Probabilistic ...
1
vote
3answers
26 views

Solving for n in the exponent.

Well, it's another question I feel like I should know. I'm trying to model the number of successes before the first failure. The probability of successes is given as $p$, which makes the probability ...
4
votes
1answer
276 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 ...
0
votes
0answers
3 views

Maximum diagonal entry of a multivariate normal sample covariance matrix

Let $\Sigma$ be a covariance matrix of $n$ data points in $\mathbb{R}^p$. So $\Sigma$ is $p\times p$. Suppose that the $n$ points are drawn from the distribution ...
26
votes
1answer
477 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
1
vote
0answers
17 views

Simple Question about Almost Sure Convergence

If I can show for some event $A_n$, $$ \sum_n P(A_n) < \infty. $$ Then by First Borel-Cantelli Lemma, I get $P(A_n \; i.o.) = 0$; But I still confused about how this infinitely often connects to ...
0
votes
1answer
8 views

A property of conditional expectation

Given a probability space $(X , \mathcal{M} , m)$ and $\mathcal{A}$ is a $\sigma$ sub algebra of $\mathcal{M}$. Let $\mathbb{E}$ be the condition expectation given $\mathcal{A}$. Given $f$ is an ...
1
vote
1answer
17 views

Combinations of colored cubes

Given three cubes and three colors (say red, green, and blue) I would like to calculate how many different ways I may associate colors with the cubes, not counting options like BGG and GBG twice. I ...
0
votes
1answer
25 views

Bayesian network and unknown probability

I'm trying to solve questions regarding bayesian network, and now I was wondering if it is possible to know the probability of an unknown variable in the tree. For instance, I have this tree, ...
0
votes
3answers
43 views

catch fish problem

There are 20 fish in a lake, among them 5 are trouts. What is the probability that a fisherman have to catch more than 6 fish in order to obtain 3 trouts? Assume he keeps every fish he caught. How ...
0
votes
1answer
17 views

Negative binomial with conditional probability

Let X be a random variable that follows a negative binomial distribution: NB(r=4, p=0.4) Calculate P(X = 8 | x > 6) I know how to calculate P(X = 8): $$ \binom{7}{3} \cdot (1 - 0.4)^{7-3} \cdot ...
-2
votes
2answers
25 views

probability question [on hold]

when rolled a fair dice 50 times what is the probability of rolling a 3 exactly 18 times
1
vote
0answers
6 views

Number of samples with replacement to reach expected coverage of population under non-uniform sampling

I am interested in finding the number of times $n$ I need to draw with replacement from a population of size $N$ such that the expected proportion of the population seen is at least $P$. From this ...
1
vote
1answer
24 views

Venn- Diagrams, Probability

I want to know how to draw a Venn Diagram with the given information below.. There are 30 students: 16 are girls; There are 7 girls and 6 boys who have blue eyes. A student is randomly ...
1
vote
0answers
32 views

Intuition for problem involving binomial random variable

Question: The below algebraic solution is simple enough. But is there a way to "see" the answer using a clever trick or intuition? Given the algebraic solution, I feel like there should be. I just ...
1
vote
1answer
21 views

Weak convergence of random variables implies $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$

Proof that, if $X_n \rightarrow X$ weakly, then $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$. I know, that I should use Fatou's lemma but I don't know what can I do first.
2
votes
1answer
11 views

Likelyhood of Poisson Distribution

The number of accidents in a week follows a poisson distribution with mean $\lambda$. Likelyhood is given as $$L(\lambda)=\frac{ \lambda^{\sum_1^n x_i } e^{-n\lambda}} { \prod x_i!}$$ However only ...
10
votes
3answers
116 views

Couple Probability

The problem states that there are 12 boys and 12 girls. Each boy chooses a girl at random and each girl chooses a boy at random. If a boy and a girl choose each other, they form a couple. It then asks ...
1
vote
2answers
29 views

Convergence in probability realated question

Consider $X_n$ and $Y_n$ be two real-valued random sequences, if $$P(X_n \neq Y_n) \rightarrow 0 \text{ as $n \rightarrow \infty$}$$ is it equivalent to say that $X_n$ converges to $Y_n$ in ...
0
votes
2answers
25 views

What is the probability that the engines will allow a safe landing?

Each of the four engines of an airplane are functioning corectly on a given flight with probability of 0.99, and the engines function independently of each other. Assume that the plane can make a safe ...
-1
votes
1answer
16 views

A joint pdf question [on hold]

I need help over a question. I appreciate all helps.Thank you.
0
votes
0answers
12 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
-1
votes
0answers
36 views

Derive the expected value of $X^{0.5}$ [on hold]

I am doing a question considering a continuous random variable X and have calculated the expected value and variance from the probability density function given. I am unsure of what the expected value ...
0
votes
1answer
371 views

expected value of maximum function.

Suppose a fair coin is tossed four times. Let X be the number of heads obtained. Comute the expected value of max{X, n − X}. Actually I cannot understand what the problems means. can you help me to ...
2
votes
1answer
61 views
+100

A Law of Large Numbers Without Replacement

Let $(n_1,...,n_r)$ be $r$ positive integers, and let $n=n_1+...+n_r$. Fo each positive integer $m$ consider an urn containing $mn$ balls, of which $mn_1$ are of type 1,..., $mn_r$ of type r. For each ...
1
vote
0answers
30 views
+50

Aumann-Shapley Uniformly Better Principle

Let $n_1,..,n_r$ be $r$ positive integers, and let $1 \leq k \leq n$, where $n=n_1+...+n_r$. Consider an urn containing $r$ different types of balls, $n_1$ balls of type 1, $n_2$ balls of type ...
1
vote
0answers
13 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
2
votes
1answer
25 views

Probability that an event will occur X times in a row at any point in Y trials?

Event AA has a $60\text{%}$ failure rate. Given $256$ trials, what is the probability that at some point event AA will fail $9$ times in a row? Is there a formula that is fairly plug-and-play? I ...
0
votes
0answers
31 views

Understanding what $P - P \log(P)$ means for an event of probability $P$

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space, $X$ be a random variable, and $E \in \Sigma$ be an event with $\mathbb{P}(E) = P$. Then $P - P \log(P) \in [0, 1]$, for all $P \in (0, 1]$, ...
1
vote
2answers
45 views

Show that $P(X=c)=1 $for some constant c

Suppose $X$ and $Y$ are independent random variables, also $X$ and $X-Y$ are independent. Prove that $$P(X=c)=1$$ for some constant c. I tried using moment generating function, please give me some ...
0
votes
1answer
19 views

Does $E(|X_n - X|) \rightarrow 0$ implies $X_n$ converges in probability to $X$?

I think it does, I've tried proving it by using Chebishev's Inequality but it only prove that it works with quadratic convergence and I can't adapt it... Can you help me please? Thank you very much! ...
0
votes
0answers
10 views

Expected Probability of a Random Agent and a Probabilistic Agent

I'm running simulations on two agents: random agent and probabilistic agent. The world they are running in is the Wumpus World where the agent is dropped in a 4x4 grid where each cell has a 20% chance ...
1
vote
0answers
25 views

A probabilty of error calculation

Let's assume I have $N$ binary strings $\{T_1,T_2,\ldots,T_N\}$ of length $L$. All these strings satisfy a minimum hamming distance with respect to a reference binary string R with $\|R\|_1$ ones and ...
0
votes
0answers
17 views

Strong law of large numbers when sample size is a random variable

For a sequence $X_1, X_2, \ldots, X_n$ of i.i.d. random variables with mean $\mu$, the strong law of large numbers tells us that $$\sum_{i=1}^{n} \frac {X_i} {n} \xrightarrow{a.s.}\ \mu ...
0
votes
0answers
14 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
0
votes
0answers
16 views

Find inequality for gaussian density

Let $C>0$ be a fixed constant. Is it true that $$Cx^2 e^{-x^2}\leq e^{-\frac{x^2}{C}}?$$ More generally, if we have a power $x^p$ in front of the exponential, do we have that $$(C^{1/2}x)^p\leq ...
0
votes
0answers
40 views

A probability of error calculation

Let's assume I have N binary strings {$T_1$,$T_2$,...$T_N$) of length L. All these strings satisfy a minimum hamming distance with respect to a reference binary string R with $||R||_1$ ones and the ...
0
votes
1answer
11 views

Probability of getting an outlier in a normal distribution

Given $ N $ data points that fit a normal distribution, what is the probability that the $ N+1^{th} $ data point is further away from the mean of the distribution than the previous $ N $ data points?
3
votes
2answers
101 views

Probability that each bucket has $\geq 3$ balls

There are $30$ buckets. John throws $20$ balls, each time landing uniformly among the buckets. What is the probability that no bucket contains $\geq 3$ balls? If the question were $\geq 2$ balls, we ...
0
votes
0answers
16 views

In Northern Yellowstone Lake, earthquakes occur at a mean rate of 1.3 quakes per year. Let X be the number of quakes in a given year. [on hold]

In Northern Yellowstone Lake, earthquakes occur at a mean rate of 1.3 quakes per year. Let X be the number of quakes in a given year. (a) Justify the use of the Poisson model. ...
0
votes
0answers
20 views

recognize the distribution corresponding to this characteristic function

The characteristic function of a random variable X is given as $$\frac{3+cos(t)+cos(2t)}{5} $$, what is the distribution of X? I was thinking of the discrete random variable X=,0,1,2 with mass ...
-2
votes
1answer
21 views

What is the probability that all are eligible? What is the probability that at least one is ineligible? [on hold]

Past insurance company audits have found that 2 percent of dependents claimed on an employee’s health insurance actually are ineligible for health benefits. An auditor examines a random sample of 10 ...
-1
votes
0answers
22 views

What is the right probability?

Three men, Abel, Baker and Charlie, are in jail in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The guard knows which one is pardoned, but is ...