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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2answers
197 views

Characterisation of conditional independence

Here is a problem I couldn't solve. Given a probability space $ (\Omega, \mathcal{A},\mathbb{P}) $, and $ \mathcal{F}, \mathcal{G}, \mathcal{B} $ sub-$\sigma$-algebras of $\mathcal{A}$. Is it ...
1
vote
1answer
81 views

Asymptotic equivalent of the law of lotto minimal value

This question is inspired by this one, where the law of the minimum $X$ of $m$ elements sampled without replacement from $\{1, \dots, n\}$ was investigated. In this question we wrote that the number ...
0
votes
1answer
71 views

Bound of absolute of random variable

If I know the $\mathbb{E}(X)$ and $\mathbb{E}(X^2)$ of some random variable $X$, can I get $\mathbb{E}(|X|)$? Or are there any bounds related to $\mathbb{E}(|X|)$?
0
votes
3answers
2k views

Probability of winning a 7 game series in 6 games

Team A and B are playing a best of 7 series, with the first team to win in 4 games winning the series. Team A has the probability $\dfrac{1}{2}$ of winning a game. If the series lasts 6 games, what ...
-3
votes
1answer
181 views

A question in Probability, resignations from stores

"Stores A,B and C have 50,75 and 100 employees and respectively, 30, 60 and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns ...
1
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2answers
244 views

A question in Probability, aces drawn from two halves of a shuffled deck

"A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves, it turns out to be an ace. The ace is then placed in the second half-deck. The ...
35
votes
9answers
4k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
0
votes
1answer
81 views

Is it possible to split a single sample from a discrete uniform distribution into two samples from two smaller distributions?

Suppose I have a single integer sample $k$ from a discrete uniform distribution such that $0 \le k \lt 2^{32}$. Is it always possible to interpret this sample as a pair of samples $m, n$ from two ...
0
votes
2answers
2k views

Validity of a Probability Density Function [duplicate]

Possible Duplicate: Probability Density Function Validity If $X$ is a continuous random variable with range $[x_l,\infty)$ and p.d.f. $f_x(X)\propto x^{-a}$, for $x\in[x_l,\infty)$ for ...
0
votes
3answers
140 views

From a mathematical point of view is it optimal in no limit texas hold em to play with more money than less?

I noticed the other time a friend of mine went to a casino and bought in 100 dollars for a 1-2 table. Other players had heavier buy ins. I have received two opposing arguments. One says that buying in ...
0
votes
0answers
185 views

Relationship between a.e. uniqueness of conditional expectations and a.e. differentiability of distribution functions?

Suppose $(X,Y)$ are random variables that are continuously distributed. Conditional probabilities are just conditional expectations: $P[Y \leq y | X = x] = E[1(Y \leq y) | X = x]$, and conditional ...
1
vote
2answers
1k views

Probability Density Function Validity

If X is a continuous random variable with range $[x_l,\infty)$ and p.d.f. $f_x(X) \propto x^{-a}$, for $x\in[x_l,\infty)$ for some values $x_l > 0$ and $a \in \mathbb{R}$. How do I calculate the ...
2
votes
1answer
2k views

Implementation of the Baum-Welch algorithm for HMM parameter estimation

In order to learn HMM thoroughly, I am implementing (in Matlab) the various algorithms for the basic questions of HMM. I've implemented the Viterbi, posterior-decoding, and the forward-backward ...
1
vote
2answers
144 views

Question on the standard normal distribution.

Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find $$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
1
vote
2answers
85 views

Calculate the probability of determined events.

Sorry if this is an easy question but I've only basic Math skills :) I have 3 people. They could say Yes or No, dependent events. So I have 8 possible scenarios: ...
1
vote
2answers
241 views

Conditional probability given that $X$ and $Y$ are uniform

Suppose that $X$ and $Y$ are uniformly distributed on $0\lt |x| + |y| \lt 1$. How do I find $P(Y\gt 1/4 ~\vert ~X=1/2)?$
4
votes
2answers
398 views

How do we justify certain steps in the solution to gambler's ruin?

The problem of gambler's ruin asks the following: suppose a player begins with $k$ units of money, $0<k<N$. Each turn he flips a coin and either gains a unit of money with probability $p$ or ...
1
vote
1answer
348 views

sum of independent variables

Suppose you have a sum of IID random variables (uniformly distributed in [0, 1]) $$S = \sum_{i=1}^N X_i$$ if I want to have a rough idea of the average value of $N$ such that the sum is equal to some ...
4
votes
1answer
2k views

How to calculate the expectation of $XY$?

Suppose I am given the joint pdf of $X$, $Y$, and I am asked to find the $\operatorname{cov}(X,Y)$. I know that $\operatorname{cov}(X,Y)=E(XY)-E(X)E(Y)$ and I know how to find $E(X)$ and $E(Y)$. My ...
1
vote
1answer
304 views

Questions about Bayesian Inference Scenario

Can someone help me with the following scenario, found on the Wikipedia page on Bayesian Inference: Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, ...
1
vote
5answers
600 views

How to explain this paradox involving coin-tosses?

I do this experiment: I flip fair coin, if it comes heads on first toss I win. If it comes tails, I flip it two times more and if both heads I win. Else, I flip it 3 more times, if it comes heads all ...
1
vote
2answers
81 views

Question about implication and probablity

Let $A, B$ be two event. My question is as follows: Will the following relation holds: $$A \to B \Rightarrow \Pr(A) \le\Pr(B) $$ And why?
182
votes
5answers
35k views

In Russian roulette, is it best to go first?

Assume that we are playing a game of Russian roulette (6 chambers). Assume that there is no shuffling after the shot is fired. I was wondering if you have an advantage in going first? If so, how big ...
1
vote
3answers
215 views

Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
1
vote
1answer
88 views

Chance of selecting a duplicate?

When you select things at random repeatedly (with replacement or whatever) out of a field of N possible things, how do you calculate the probability that something has been chosen X times after Y ...
0
votes
1answer
2k views

Poisson process and probability phone calls

If the phone calls from a central are made in a Poisson process ( N(t), t≥ 0 ), in average every 10 minutes they have one phone call. calculate the probability that no call is received in the range ( ...
0
votes
1answer
145 views

Integration about standard normal

Let $N(x)$ denote the cdf of standard normal and $n(x)$ denote the pdf of standard normal. How to evaluate the integral $\int\limits_{-\infty}^\infty N(a+x) n(x) \mathrm{d} x$ ? Thanks a lot!
2
votes
3answers
114 views

Conditional Normal

$X,Y,Z$ are standard normal R.V. What is the value of $\operatorname{E}[X|X+Y+Z=1]$ and $\operatorname{Var}(X|X+Y+Z=1)$? I think the first one should be $1/3$ by symmetry but don't know how to ...
3
votes
3answers
903 views

Probability of an odd number in 10/20 lotto

Say you have a lotto game 10/20, which means that 10 balls are drawn from 20. How can I calculate what are the odds that the lowest drawn number is odd (and also how can I calculate the odds if it's ...
-1
votes
1answer
89 views

Calculating the probability

Assume $T_1,T_2...T_{300}$ are the time one talking in the lesson before he was stopped by his teacher.$T$ are $iid$ and follow an exponential distribution with mean $exp({\theta})$ what is the ...
14
votes
3answers
1k views

Odds of guessing suit from a deck of cards, with perfect memory

While teaching my daughter why drawing to an inside straight is almost always a bad idea, we stumbled upon what I think is a far more difficult problem: You have a standard 52-card deck with 4 suits ...
3
votes
2answers
612 views

Random walk probability/expected value

With what probability, starting at node $g$, does node $d$ get hit before node $e$ in the graph below? What is the expected value of number of steps you need to hit $\{d,e\}$ (at least one of them) ...
0
votes
0answers
56 views

Probability of random like permutation

Consider a permutation $S$ of $n$ numbers between 1 to $n$. We know probability $P(S[i]=j)=a_{ij}$ for $1\leq i,j \leq n$. We want to find $P(S[4]=6 | S[30]=25)$. We can use approximations like ...
5
votes
1answer
145 views

area ratios when random line cuts regular polygon

A point is randomly selected on one side of a polygon, and another point is randomly selected on one of the other sides. A line is drawn through those points. What is the mean expected ratio of the ...
1
vote
2answers
94 views

about the percentile of $f(X)$

Is the, say $90\%$, percentile of $X$ the same as the $90\%$ percentile of $aX+b$, where $a, b$ are constant? I mean, to calculate the $90\%$ percentile of $X$, can I use the central limit law to ...
3
votes
1answer
96 views

$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
1
vote
0answers
136 views

A question on bivariate discrete uniform.

Let $X$ and $Y$ be independent random variables, each uniformly distributed on $\{1,2,3,\ldots,11\}.$ I want to find $\mathrm{P}(X+Y=16).$ Well, the joint probability mass function of $X$ and $Y$ ...
2
votes
2answers
543 views

variance of the cdf with a jump

The problem is A random variable X has the cdf $$F(x)=\frac{x^2-2x+2}{2}\quad\text{if}\quad1\leq x<2$$ and $F(x)=0$ when $x<1$, $F(x)=1$ when $x\geq 2$. Calculate the variance of X(the ...
2
votes
1answer
304 views

Convergence of sum of a sequence of random variables

Consider $(X_{1},\ldots,X_{n})$ a sequence of random variable i.i.d such as $P(X_j=1)=P(X_j=-1)=\frac 12$ for all $j \geq 1$. Consider now the sequence $Y_{n} = \sum_{j=1}^{n} 2^{-j} X_{j}$ for all $n ...
10
votes
2answers
2k views

Expected number of tosses for two coins to achieve the same outcome for five consecutive flips

Consider two unbiased coins. Toss both until last 5 sequence outcome are same. That means we stop when output of the sequence of both are as follows: HTTHTHHTH , HHTTTHHTH. What is the expected ...
2
votes
1answer
121 views

What does it mean to select $O(k \log k / \epsilon^2)$ indices?

I'm reading [1] where some columns and rows of a matrix $A$ are selected by their leverage scores aiming to have CUR decomposition of $A$. In the paper $c$ is a value determining how many indices we ...
6
votes
2answers
385 views

Kolmogorov's maximal inequality for random number composition

The Kolmogorov's maximal inequality states that when $X_1,\dots,X_n$ are mutually independent random variables, each with finite variance. Set $S_j=X_1+\cdots+X_j, 1 \le j\le n.$ Then, for each ...
1
vote
2answers
3k views

Probability of a Union

I know that $$P\left(\bigcup_{i=1}^{n} A_i \right)$$ is the sum of of the probabilities of all the sample points that are contained in at least one of the $A_{i}$'s. This is the probability of ...
2
votes
2answers
1k views

Hitting a Target

If the probability of hitting a target is $1/5$, and 10 shots are fired independently, what is the probability of the target being hit at least twice? Let $X$ be the number of times the target is ...
0
votes
0answers
71 views

Averages and sampling

Suppose we draw a histogram of the distribution of all the possible sample averages drawn with replacement from the set $\{1,2,7,8,14,20 \}$. I know that $\frac{1+2+7+8+14+20}{6}$ has the highest ...
3
votes
1answer
495 views

probability unordered sample

Suppose we have a collection of six numbers $\{1,2,7,8,14,20 \}$. What is the probability of drawing with replacement the unordered sample $\{2,7,7,8,14,14 \}$? It seems that this probability would ...
3
votes
0answers
234 views

Intuition for Prohorov metric and metrization of weak convergence

According to Billingsly, let $P$ and $Q$ be two probability measures. Then the Prohorov metric $\pi (P,Q)$ is the infimum of those positive $ \epsilon $ for which the two inequalities $PA \le ...
3
votes
3answers
353 views

Expected smallest prime factor

For a random integer $x$ chosen uniformly between 2 and $n$, what is the expected value of the smallest prime factor of $x$ as a function of $n$? What is the behavior of the function as $n$ tends to ...
3
votes
1answer
199 views

Conditional probability of a general Markov process given by its running process

I have a question as follow: "Let $X$ be a general Markov process, $M$ is a running maximum process of $X$ and $T$ be an exponential distribution, independent of $X$. I learned that there is the ...
8
votes
4answers
3k views

Math story: Ten marriage candidates and 'greatest of all time'

I remember a story about a famous mathematician who was offered ten marriage candidates and had to pick one of them, with the condition he had to meet them in turn and propose during that meeting, ...