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
18 views

Proof that absolute value of a random variable is a random variable

Is this proof correct?: Proof: Suppose that $X$ is a random variable on a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$. Suppose $x \in \mathbb{R}$ and $x \geq 0$. Then $\{|X| \leq x\} = ...
0
votes
1answer
9 views

$X,Y \sim $iid $\operatorname{Exp}(\lambda),\ P(X \le t, X + Y > t)$?

$X$ and $Y$ are 2 identical exponentially distributed independent random variables. $X,Y \sim$ iid $\operatorname{Exp}(\lambda)$ What's the following probability? $$P(X \le t, X + Y > t)$$ I ...
0
votes
0answers
27 views

Number of rewards before death

I have a question regarding Poisson events with death. Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with ...
0
votes
1answer
38 views

Expectation of a symmetric function about zero

I understand that the median of a symmetric function about zero is zero because 50% of the mass is contained either side of it. It is easy to prove this result for the Expectation,however I do not ...
1
vote
1answer
34 views

Quantum probability and quantum measure theory

Do quantum probability and free probability mean the same thing - that is, they deal with noncommutative random variables? What about quantum measure theory? Is quantum measure theory the foundation ...
1
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1answer
17 views

Conditional expectation of iid nonnegative random variables

I am studying Ross's book, stochastic processes. There is the following lemma: Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then, $E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] = ...
1
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1answer
25 views

How can we measure the accuracy of prediction algorithm?

We have created a prediction algorithm, which predicts the chances of confirmation of ticket based on some parameters, and gives the prediction in percent. Now, how do I measure how close the ...
0
votes
1answer
17 views

What is the probability of obtaining the right, left, Ace, King, and Queen of trump in a 5-card hand in euchre?

Take into account that in order for that hand to be the best hand, the dealer must also flip over a card that matches the suit that your best hand is for. (If you have the J of spades, Jack of clubs, ...
1
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1answer
12 views

Bivariate normal distribution of points

I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this: I found multiple terms for describing a "circular normal distribution" and yet, I'm not ...
16
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5answers
1k views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
-1
votes
1answer
19 views

How many different hands are possible if two spades are drawn from a deck of cards? [on hold]

Two cards are drawn from a deck of 52 cards. If both cards are spades, how many different hands are possible? (Note: A hand of cards is the collection of cards a player is holding, a standard deck ...
2
votes
1answer
15 views

Probabilistic Conditioning. Please help me finish the solution to the problem.

I have to solve the following problem: The real random variables $X$ and $Y$ are independent and have a uniform distribution $U([0,1])$. Find $$\mathbb{E}\left( \frac{3 X-Y+1}{\sqrt{X+Y+1}} | \quad ...
0
votes
1answer
431 views

Greatest of three random variables

Assume that we have $3$ not equal random variables $(A, B, C)$. If we know that $$Pr(A>B)=x, \quad Pr(A>C)=y, \quad Pr(B>C)=z$$ What is $Pr(A$ is the greatest one)? I know that $Pr(A$ is ...
1
vote
2answers
50 views

$X$ and $Y$ are independent Poisson$(\lambda)$, $\lambda\sim\mathrm{exp}(\theta)$. What is the conditional distribution for $X$ given that $X+Y=n$?

To clarify, the parameter $\lambda$ is a random variable with exponential distribution and parameter $\theta$. Can someone please tell me if I've correctly computed the conditional distribution for ...
1
vote
1answer
521 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
4
votes
2answers
63 views

Confusing probability problems based on product rule and combinations

I am going thru probability exercise. Faced first problem: Book Q1. Ten tickets are numbered 1,2,3,...,10. Six tickets are selected at random one at a time with replacement. What is the ...
1
vote
1answer
19 views

How do i calculate the probability of the relay in the circuits?

I am trying to solve my following probability question but i can't see how to make any progress. Any help will be highly appreciated Question: The probability of the closing of the i-th relay in the ...
3
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3answers
3k views

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

I have this question as an example in my maths school book. The solution given there is:- E = the man reports six P(S1)= Probability that six actually occurs = $\frac{1}{6}$ P(S2)= Probability ...
-3
votes
0answers
24 views

a question about sample space [on hold]

How to represent the following statement mathematically: "The event $\{A_n \text{occurs infinitely often} \}$ is $\{ \omega \in \Omega | \omega \in A_n \ \text{for infinitely many values of} \ n ...
0
votes
0answers
25 views

Identifying and separating two different distributions in a set of mixed data

Data The data at hand comprises distances between successive points of known location, which occur with set limits (red line is of finite, known length): Points are chosen in succession, one after ...
8
votes
1answer
44 views

Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
1
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3answers
23 views

Probability - consecutive numbers

Question: Three numbers are selected out of the first 30 natural numbers. What is the probability that none of them are consecutive? I know that the total possibilities will be $^{30}C_3$ ...
0
votes
1answer
668 views

Find the mean and variance of the total service time using the Poisson distribution

Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. If it takes approximately ten minutes to serve each customer, find ...
0
votes
1answer
37 views

poisson distribution probability problem

I am working on a Poisson distribution problem stated in the main question and got stuck and do not know how to proceed as I did not understand the next question on how to work it out The following ...
0
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2answers
35 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
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votes
1answer
26 views

If $w$ is a discrete random variable then is $P(w|x)$ a density or mass function? [on hold]

$w$ is a discrete random variable. $x$ is a continuous random variable. Then should I denote $P(w|x)$ as a probability density function or probability mass function, and why ?
0
votes
1answer
25 views

N balls having M different colors in a box, how many times do I need to pick to get one particular color?

There are $N$ balls of $M$ different colors in a box i.e $c_1$ balls of color $1$ and so on. $c_1 + c_2 + \dots + c_M=N$, $c_1, c_2, \dots, c_M$ are known. We are looking for a ball of a particular ...
8
votes
2answers
555 views

Bingo Probability Problem

A Bingo card has 25 squares with numbers on 24 of them, the center being a free square. The integers that are placed on the Bingo card are selected randomly and without replacement from 1 to 75, ...
1
vote
0answers
11 views

Estimating the expectation of a derivative [on hold]

Assume $Y$ is a continuously differential function of $X$. Given i.i.d. data $(x_i,y_i)_{i=1}^n$, I would like to estimate $E\left[\left.\frac{\partial Y}{\partial X}\right|_{X=X_0}\right]$. ...
1
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0answers
32 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
1
vote
1answer
452 views

What is number of perfect matchings in a bipartite graph

Let's $G=(U,V,E)$ be a random balanced Bipartite graph graph which $|U|=|V|=n$. What is the number of random graphs that has a perfect matching? I think that the number of possible graphs is ...
0
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0answers
25 views

Probability and Expected number of games played

I am wondering how I would I apply Markov chains or martingales to solve problems of the following type: Example : Two players play games against each other until either of them wins 3 games in a ...
1
vote
1answer
50 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
-1
votes
1answer
17 views

Probabilities for selections from a set

This seems like it would be a common question, but I can't come up with a search that provides an answer to my question, so I'm asking it here. I have a set of unique numbers ...
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2answers
31 views

Having an independent event with animals

In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments must have cat and dog for events "have dog" and " have ...
6
votes
3answers
53 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
-1
votes
1answer
33 views

Probability - conditional

The probability that bulbs are detected faulty if they are defective is 0.95 and the probability that bulbs are declared fine if in fact they are fine is 0.97. If 0.05 of the bulbs are faulty, what is ...
2
votes
1answer
32 views

How to compute the expected value of one random variable over sum of iid random variable

If $X_1,\ldots,X_n$ are independent identically distributed positive random variables, prove that $E(\frac{X_i}{X_1+\cdots+X_n})=\frac{1}{n}$, $i=1,\ldots,n$. Can someone give me a hint?
1
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2answers
40 views

Random walk in one dimension with different probabilities

As the title suggests, I'm concerned with a typical random walk problem, where the probability to go right is $p$ and the probability to go left is $q=1-p$. I was trying to find the probability of ...
0
votes
0answers
32 views

The Gambler's Ruin without using random walks

This is more of a doubt. I understand that this problem can be described with Markov chains and the recursion solved without much trouble. However I've seen that some people casually say that $$ ...
-2
votes
1answer
26 views

Probable winner of last coin game of a series, where winner from one game has disadvantage the next game?

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner is the first person to obtain a head. They play this game several times, with the stipulation that the loser ...
-5
votes
1answer
32 views

Expected Value and Expected Variance Probability [on hold]

Suppose a production line manufactures ball-bearings with a radius that is uniformly distributed between 1.8mm and 2.2mm. What is the probability of (a) the expected value of the volume, and (b) the ...
1
vote
1answer
50 views

Probability of no 6 or no 5 when dice is rolled n times

Can anyone guide me in the general direction of the answer to the following: A die is rolled $n$ times $$A = \text{no $6$s}$$ $$B = \text{no $5$s}$$ $$P(A\cup B) = \;?$$ I am first finding $P(A)$ ...
0
votes
1answer
25 views

probability distribution of the winning amount

Be A_n the event that a worker spends to process certain component with probabilities according to the table below: For each piece processed, the worker earns a fixed US 2.00, but if he processes ...
0
votes
2answers
44 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...
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votes
0answers
24 views

Probability, normal distribution, car collision [on hold]

There is a question in the book Principles of Statistics by M.G. Bulmer which I'm stuck on. Here goes: The reaction time of two motorists A and B are such that their braking distances from 30 m.p.h. ...
-1
votes
1answer
37 views

What is a nice, clean proof to show that a fair coin toss satisfies axioms of probability?

If we assume H=Heads T=Tails and we're dealing with a fair coin what is a good way we can show that Kolmogorov Axiom has been satisfied?
-1
votes
0answers
15 views

Expected Value of covariance [on hold]

You have an urn with balls that are either red or blue, and striped or not. What is the probability distribution that describes the number of blue balls drawn given the number of striped ones drawn? ...
0
votes
0answers
58 views

From the binomial distribution

A single cell can either die, with probability $0.1$, or split into two cells, with probability $0.9$, producing a new generation of cells. Each cell in the new generation dies or splits into two ...
1
vote
1answer
50 views

The ant is moving through the coordinate system, Started at $(0,0)$ to $(4,4)$. What is the probability that the ant will find food at $(3,2)$?

The path to the $(3,2)$ is $3+2 \choose 3$ or $3+2 \choose 2$. Total path is $4+4 \choose 4$ And the probability is : $ \frac{3+2 \choose 3}{4+4 \choose 4}$ = $ \frac{5 \choose 3}{8 \choose 4}$ = ...