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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1answer
203 views

probability that two people share the birthday and the date of death

I've encountered an interesting probability problem that my little amount of knowledge does not help me to solve. Select 1000 people from the past, and ignore the year. Also assume that the 365 days ...
2
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0answers
118 views

Is it possible to calculate the balanced cost of parameters' increase in card game? How?

I wonder if it's possible to calculate the balanced cost of parameters' increase for the card game. Game rules: Each player draw 7 cards at the beginning of the game and then one card each turn (...
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1answer
47 views

Volume and Probability of a region given by a random variable

I am currently reading this paper. It is about nearest neighbors of a query point $X_q\in\mathbb{R}^k$ within a point set $P=\{X_i\mid X_i\in\mathbb{R}^k\}$, where the points have distribution $p(X)$ ...
2
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0answers
64 views

Poker dice probability of rolling 2 pairs

Poker dice are played by rolling 5 dice. Let A be the event of rolling 2 pairs. (e.g. 1,1,2,2,3.). Find $\mathbb{P}(A)$. So my answer is as follows: $$\mathbb{P}(A) = \frac{\binom{6}{2}\binom{5}{2}\...
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0answers
59 views

The probability of the element at index $i$ is greater than all following items

Consider an array which is a permutation of $[1,2,3,\dotsc,n]$. Each of the permutation is equally likely (Uniform distribution). What is the probability that the item at index $i$ is greater than ...
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1answer
101 views

2 identical candies are given to 2 distinct children at random. What is the Probability that each children has at least one candy?

OK i have two different ways to approach the problem and both give different answers, it would be great if someone could clarify which of them is correct(if they are). Method 1: Since the candies are ...
2
votes
1answer
27 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other words,...
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2answers
2k views

Conditional probability rolling two dice

Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? I already know the answer, but am having some trouble ...
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2answers
33 views

Equivalence involving expectation

I am stuck with the following problem, where I am asked to prove/disprove the following hypothesis: Is $\mathrm{E}\{e^{\max_i X_i}\} = \mathrm{E}\{\max\limits_i e^{X_i}\}$, where the $X_i$'s are ...
1
vote
1answer
252 views

What's the probability that 2 of 4 people were born in the same month

I have this problem to solve: "Calculate what's the probability that in the four-person family at least 2 people were born in the same month". I know that I can calculate opposite event in very ...
0
votes
1answer
43 views

Expected value for random walk

A point starts at the origin and can randomly go up, down, left, right (equally likely). The question asks to write the expression of the point's position in terms of $x_1$ -units up, $x_2$ -units ...
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3answers
581 views

A box contains 10 balls numbered from 1 to 10

An urn contains balls numbered $1, \ldots, 10$. Five balls are drawn without replacement. What is the probability that the second largest of the five numbers drawn will be 8? I believe the number of ...
1
vote
3answers
43 views

formula to produce a set of probability distributions for a set of integers between a lower and upper bound with a given mean value

The goal is to establish a set of probabilities to be used to select an integer value where the probability of selecting I is Q, I+1 is R, I+2 is S, ... I+n is Z and such that the integer with the ...
0
votes
2answers
151 views

No pairs when drawing cards from deck

Suppose we are dealt five cards from an ordinary 52-card deck. What is the probability that we get no pairs (i.e. all cards are different values). I'm not sure if I've got the right answer on this ...
1
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0answers
95 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
0
votes
1answer
48 views

Confusion over Multivariate Hypergeometric Distribution

I'm reading Freund's statistics book and he says the following: "Just as the hypergeometric distribution takes the place of the binomial distribution for sampling without replacement, there also ...
1
vote
1answer
30 views

An inequality involving the normal distribution

Let $X$ be normal random variable with parameters $\mu = 0$ and $\sigma^2 =1 $. Is it true that $$ P( X > x ) \leq \frac{1}{x \sqrt{2 \pi}} e^{ -x^2/2} \text{ for }x>0 \text{ ??}$$
1
vote
1answer
78 views

Distribution of maximum of correlated Gaussians

Let $X_1,X_2,...,X_n$ be iid standard Gaussian random variables. Consider the set of random variables $M =\left\{\left( X_i-X_j\right) :i,j = \left\{1,2,\dots,n\right\} \& i\ne j\right\}$. I ...
0
votes
2answers
57 views

Show that A is an event

Let $X_n$ be a sequence of random variables on a probability space $(\Omega, F, P)$. Let's define: $$A=\{ w \in \Omega: \lim_{n \to \infty} X_n(w) \space \text{exists}\} .$$ Now, I need to show ...
1
vote
1answer
29 views

Probability Density of Mapped $y=ax^2$ to Normal Distribution

Let $x$ be a scalar random variable and let $y$ be a scalar quantity such that $y = ax^2$. If $$p(x) = \frac 1 {\sigma\sqrt{2\pi}}\, e^{-x^2/(2\sigma^2)}$$ a) Find the probability density of $y$ b)...
2
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0answers
67 views

Law of large numbers for a continuum of random variables

Consider a continuum of random variables such that each takes the value $1$ with probability $p$ and $0$ with probability $1-p$. The random variables should be essentially pairwise independent. Sun ...
0
votes
1answer
82 views

can you minus one to ge inverse conditional independence of an intersection

A patient would like to take a test to determine if he has a nasty disease. Let the variable A denote that the patient has the disease and the variable B denote a positive test,Consider a second test ...
0
votes
2answers
38 views

Pdf of scaled Nakagami?

I am trying to find the probability distribution function (pdf) of the following $$Y=a X $$ Given that $$X \sim \operatorname{Nakagami}(m,1)$$ $$a \,\text{positive constant}$$ Is the pdf of $Y$ $$...
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2answers
58 views

Mixed Poisson Distriubtion

Question: The number of accidents follows a Poisson distribution with mean 12. Each accident generates 1, 2 or 3 claimants with probabilities 1/6,1/3,1/2, respectively. Calculate the probability that ...
5
votes
1answer
493 views

Fubini's theorem for conditional expectations

I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ is finite then: $$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$ I just ...
0
votes
1answer
44 views

Poisson Distribution with two different expected value

Question: Suppose there are number of accidents at location $A$ and location $B$ each day with Poisson Distribution with rate $0.001$ and the number of patients during each accident is Poisson ...
0
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1answer
40 views

Expected length of shortest interval containing numbers drawn at random

A random idea: If you draw $n$ numbers uniformly at random from $[0,1]$, what is the expected length $L_n$ of the shortest interval that contains all but one of them? Clearly, we have $$L_2 ...
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votes
2answers
88 views

Arriving at the formula for expected value of a random variable

I was trying to solve this programming problem but I can't solve it since I am not able to arrive at the right formula for it. I dont want the solution using Dynamic Programming. I just want someone ...
1
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1answer
54 views

Binomial Process

When defining a binomial process two of the conditions that define it are: The probability of a success, p, must be fixed throughout the trials The trials must be mutually independent. What is the ...
2
votes
1answer
60 views

Let $X$ be a random variable with continuous CDF $F$. Find CDF for $|X|$ and (if $F\in C^1$) density $f_{|X|}$

Let $X$ be a random variable with continuous distribution function $F$. Find the distribution function $F_{|X|}$ for $|X|$. Suppose that $F\in C^1$ . Find the density for $|X|$. My approach was ...
0
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1answer
93 views

For each combination of k from N items. Say “Yes”, with probability p. How?

I'm considering two ways of generating a random instance of the Yes's:- (a) For each combination of k items from N: ...
0
votes
1answer
31 views

Minimize loss in -EV gamble.

Betting related terms: Multiple: a bet you make on two or more events where you win only if you guessed right all the outcomes of the events. Odds: will be expressed in decimal, eg. in a bet at odds ...
1
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1answer
31 views

Exponen distribution

The probability that a certain machine breaks during the first 370 hours of its use is 50%. How many hours may the machine be used until it breaks with a probability, which at least 85%? I don't know ...
4
votes
1answer
93 views

Probabilistic method: vertex disjoint cycles in digraphs

Let us say that a di-graph is $k$-regular if every vertex has precisely $k$ out-edges. The following theorem appears in a book I am currently studying Theorem. Every $k$-regular graph $D$ has a ...
1
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2answers
136 views

How to calculate E(X|X+Y=a) for some given a?

Suppose $X \sim exp(\lambda_1)$ and $Y\sim exp(\lambda_2)$. Than how to calculate $E(X | X+Y=a)$ for a given $a>0$. My try: $E(X | X+Y=a)=\int_0^\infty xP(X| X=a-Y)dx$ Then how to calculate $P(...
0
votes
2answers
204 views

What is the probability of three cards drawn containing no Queens, but at least one Ace or one King?

We have a standard $52$-card deck. What is the probability of three cards drawn containing no Queens, but at least one Ace or one King? The total no of draws: $\binom{52}{3}$ no. of draws without ...
0
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2answers
44 views

Probability of $n$ successes in the first $k$ trials given that there were $n+1$ successes in the total of trials

I'm having trouble with the following problem: A man found that $3$ out of $10$ inspected bottles were defective. What is the probability that the $2$ first defective bottles were found in the first $...
0
votes
1answer
340 views

How many distinct arrangements of four letters (without repeats) from the set {A,B,C,D,E} are possible?

Again the set is {A,B,C,D,E} and how many arrangment of four without repeats are possible? I'm not entirely sure how to work this problem out. Perhaps there is a formula which could be used? Thanks
0
votes
3answers
106 views

Basic probability: Is event space in probability poorly defined?

I am in a probability class and we are just getting into random variables. A random variable to me is a function that maps event to some real number. But the concept of event is difficult to swallow. ...
2
votes
2answers
51 views

Forming 4 teams of 15 players out of a set of 60

Why isn't it just $${60\choose 15} \times {45\choose 15} \times {30\choose 15} \times {15\choose 15}$$ I really don't get this problem.
2
votes
1answer
99 views

Combinatorics question with locks.

I have a question regarding the following problem. A crime detective is able to dust a lock for finger prints and he determines that the numbers: 3,4,5 and 8 have been pressed repeatedly and the ...
0
votes
1answer
43 views

Covariance of random variable as a function of distribution of noise

Consider the following stochastic difference equation \begin{equation} x(t+1) = x(t) + \nu(t+1) \end{equation} where, $x(t)\in\mathrm{R}$ be one dimensional and $\nu(t)$ be the disturbance with an ...
2
votes
1answer
184 views

Meaning of expected value?

Given probability space $(\Omega, \mathcal{F}, P)$ and $P$-measurable function $X: \Omega \to \mathbb{R}$ called a random variable, define the expectation of $X$ to be $E[X]=\int_\Omega X(\omega)dP(\...
0
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2answers
86 views

Intro to Probability Airplane

It turns out that my friend and I are boarding on the same plane. We are both travelling on economy class, and this plane has 10 single and 10 double (a total of 30) economy seats. Assuming the seat ...
0
votes
2answers
38 views

Expected Value Probability

(10 pts) Suppose that the distribution function of a random variable X is given by $$F(t) =\begin{cases} 0 & : t <−1 \\ 1/7 & : − 1≤t< 3 \\ 3/7 & : 3≤t<4 \\ 4/7 & : 4≤t<...
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vote
2answers
153 views

Introduction to probability Dice

We roll a fair die repeatedly until we see the number four appear and then we stop. What is the probability that we needed an even number of die rolls? For this problem I said that it would be the ...
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votes
1answer
44 views

Expectation of B(1) times stochastic integral? [closed]

I need to find the value of this expectation: $$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$ $B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is ...
0
votes
2answers
27 views

Relationship between CDF and Random Variable

Why is the distribution of a random variable uniquely determined by it CDF?
0
votes
1answer
188 views

A box has 10 red balls and 5 black balls.

Hey I am having trouble in solving this problem, I would really appreciate some help. A box has 10 red balls and 5 black balls. A ball is selected from the box. If the ball is red, it is returned to ...
2
votes
2answers
136 views

Bayes probability with unfair coin - what went wrong

A box has 1000 pennies. One penny in the box has 2 heads. A coin is selected at random and flipped 5 times. If the coin comes up heads each time, what is the probability that the selected coin had two ...