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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2answers
3k views

Calculating the probability mass function

Let $X$ be 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 $a \in \mathbb{R}$. Assume $x_l$ = 0.5. ...
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2answers
209 views

Chance for picking a series of numbers (with repetition, order doesn't matter)

I want to calculate the chance for the following situation: You throw a die 5 times. How big is the chance to get the numbers "1,2,3,3,5" if the order does not matter (i.e. 12335 = 21335 =31235 etc.)? ...
4
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1answer
329 views

A question about independent events

While studying probability, the following question arose: Let $H$ be an event and let $\mathcal{H}=\lbrace H_\lambda|\lambda\in\Lambda\rbrace$ be a family of events in probability space ...
2
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2answers
210 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 ...
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1answer
83 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 ...
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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 $\mathbb{E}(|X|)$?
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3answers
3k 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 ...
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1answer
236 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 ...
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2answers
330 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 ...
37
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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
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1answer
85 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 ...
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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 ...
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3answers
141 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 ...
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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 ...
3
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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 ...
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2answers
146 views

Question on the standard normal distribution.

Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find $$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
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2answers
90 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: ...
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2answers
269 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
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2answers
445 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
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1answer
366 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
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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 ...
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1answer
363 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, ...
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5answers
637 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 ...
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2answers
83 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?
202
votes
5answers
38k 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 ...
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3answers
228 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
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1answer
99 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
146 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
958 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 ...
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1answer
91 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
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3answers
2k 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
655 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) ...
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0answers
60 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
156 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 ...
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2answers
96 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
100 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
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0answers
152 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
660 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
320 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
122 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
395 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 ...
0
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2answers
4k 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
2k 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 ...
3
votes
1answer
624 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
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0answers
250 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
390 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
201 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 ...