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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Problem of Conditional Probability

I am learning Probability from Sheldon Ross book. One of the problems starts by giving the probability $P_N$ that there are no matches when $N$ people select from among their own $N$ hats as ...
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1answer
13 views

Expected value using indicator variables

Randomly, $k$ distinguishable balls are placed into $n$ distinguishable boxes, with all possibilities equally likely. Find the expected number of empty boxes. PROPOSED SOLUTION: Let $I_j$ be the ...
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Sufficient condition for $E(wu\mid v)=0$ given that $E(u\mid v)=0$?

I'm trying to figure out what condition concerning $w$ and $v$ would be enough for me to infer that $E(wu\mid v)=0$ given that I already know $E(u\mid v)=0$. Clearly, $w$ is a constant works: ...
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2answers
47 views

I'm not able to solve conditional probability questions!! [on hold]

You are given: $\Pr(A) = {2\over 5}$, $\Pr(A ∪ B) = {3\over 5}$, $\Pr(B\mid A) = {1\over 4}$, $\Pr(C\mid B) = {1\over 3}$, and $\Pr(C\mid A ∩ B) = {1\over 2}$. Find $\Pr(A\mid B ∩ C)$ Okay, ...
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1answer
31 views

Conditional Gambler ruin problem

A gambler repeatedly plays a game where in each round, he wins a dollar with probability 1/3 and loses a dollar with probability 2/3. His strategy is “quit when he is ahead by 2 dollars”, though some ...
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0answers
9 views

Inverse function for a sort of negative binomial distribution

I am trying to find the inverse function of $f(p) = \sum_{k=0}^{6}{\binom{6-H+k}{k} p^{7-H} (1-p)^k}$, where $0 \leq H \leq 6$ is a constant integer. Any ideas on how to do this? Or perhaps equally ...
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0answers
17 views

Quadrant probability of non-centric bivariate normal distribution

Suppose $(X,Y)$ has a bivariate normal distribuion with non-zero mean vector $\mu$ and covariance matrix $\Sigma$. What should $\mathbb{P}(X>0,Y>0)$ be? My attempt gives me an definite ...
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3answers
42 views

Gender Birth problem - Conditional probability

A family has two children. Assume that birth month is independent of gender, with boys and girls equally likely and all months equally likely, and assume that the elder child’s characteristics ...
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0answers
43 views

How many divisors of the combination of numbers?

Find the number of positive integers that are divisors of at least one of $A=10^{10}, B=15^7, C=18^{11}$ Instead of the PIE formula, I would like to use intuition. $10^{10}$ has $121$ divisors, ...
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0answers
35 views

Lottery Expected Value Question

Background: The Daily 3 game is a daily game, drawn every day except for Saturday and Sunday. It consists of three sets of balls, each numbered from $0$ through $8$ ($9$ is omitted due to its visual ...
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4answers
52 views

Is conditional probability always meaningful

Problem: A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement. Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the ...
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1answer
15 views

How to calculate the partition function of a given distribution?

As noted in A FULL BAYESIAN APPROACH FOR INVERSE PROBLEMS, let $ y = Ax + n$, where $x$ is a $m$ dimensional signal and $n$ is white Gaussian noise with precision $\beta$, so we have: $$ y|x, \beta ...
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1answer
34 views

Probability of Two People Choosing the Same Number between 1 and 50 or choosing 2 numbers that add to 50 [on hold]

If two people are asked to each choose a number between 1 and 50, what is the probability that they choose the same number, or choose numbers that add to equal 50?
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1answer
24 views

Counter example: $X$ and $Y$ normal imply $(X,Y)$ bivariate normal

I vaguely remember this construction from one of my courses: Suppose that $X\sim N(0,1)$ and $Z$ is $\pm 1$ with probability $\frac{1}{2}$ each. If $X$ and $Z$ are independent, then $Y\equiv XZ$ is ...
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2answers
19 views

Biased and fair coin in Hat flipped

Two coins are in a hat. The coins look alike, but one coin is fair (with probability 1/2 of Heads), while the other coin is biased, with probability 1/4 of Heads. One of the coins is randomly pulled ...
1
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1answer
58 views

Expected Value of a Mosquito

A mosquito is walking at random on the nonnegative number line. She starts at $1$. When she is at $0$, she always takes a step $1$ unit to the right, but, from any positive position on the line, she ...
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0answers
30 views

What is the area covered by a Random walk in a 2D grid?

I am a biologist and applying for a job, for which I need to solve this question. It is an open book test, where the internet and any other resources are fair game. Here's the question - I'm stuck on ...
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1answer
33 views

Probability of getting a five digit number divisible by 5 but with no two consecutive digits identical

A five digit number is written down at random. What is the probability of getting a number that is both divisible by 5 and doesn't have any 2 consecutive digits identical? I tried to analyse the ...
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1answer
35 views

An urn full of balls problem [on hold]

Among $10$ balls in a bag , $6$ of which are black , $4$ white , $3$ balls were removed randomly . What is the probability that from the remaining $7$ balls , if one ball is chosen at random , it is ...
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1answer
25 views

a pin is spun on a flat table [on hold]

A pin whose centre is fixed on a flat table is randomly and independently spun twice.Each time the final position is noted by drawing a line segment.what is the probability that the smallest angle ...
1
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1answer
27 views

If 3 people put their hat in a box, but the hats are mixed up. How likely is it that AT LEAST one person getting their hat back.

If 3 people put their hat in a box, but the hats are mixed up. How likely is it that AT LEAST one person gets their hat back. Consider all possibilities. Then what about 4 people. Please use ...
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3answers
25 views

Probability rolling two different sided die and sum being a number

I'm building an app (for those curious, for DnD) and I came across an issue with some math I did. I need to know the probability of rolling a certain number when there are two or more different sided ...
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0answers
32 views

boxes around a circle [on hold]

Suppose You have $10$ boxes numbered from $1,2,3,......,10$ arranged sequentially around a circle. We perform $100$ trials.At each step you have to choose a specific box with probability $\frac ...
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1answer
57 views

How and why do rumors/gossip spread? [on hold]

I should clarify what I mean by gossip (this is taken from wiki): Idle talk or rumor, especially about the personal or private affairs of others. That seems accurate enough, though alternative ...
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0answers
20 views

Regression in Bivariate Normal

Suppose $(X_i \hspace{4pt} Y_i)'$ are $i.i.d$ $N_2 (\bf{\mu,\Sigma})$, $i=1(1)n$ where $E(X)=\mu_x$, $E(Y)=\mu_y$ and $\Sigma$ is given by \begin{bmatrix} \sigma^2_x & \rho\sigma_x\sigma_y\\ ...
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0answers
15 views

Find conditional probability of a mixture model

given is the following: A mixture model comprises a non-observable $\{ 0,1\}$-valued random variable $X$ such that $P(X=1)=1-P(X=0)=\pi$ and an observable variable $Y$ such that $Y\mid X=0$ is ...
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1answer
21 views

Probability a card can win a trick with a trump suit

I'm working on the AI for a card game that uses a standard deck of 52 cards consisting of 13 cards in 4 (spades, clubs, diamonds, hearts) suits. Each player starts with 13 cards in their hand. ...
4
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4answers
128 views

What is the probability of getting a 3 or higher on a six sided die, if I reroll after failing the first time?

Just as the question says... What is the probability of rolling a 3 or higher on a six sided die, if I reroll the die a second time when I fail the first time?
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0answers
51 views

Question about conditional probability [on hold]

I am working on some exercise questions from probability. I am stuck at this question. Can somebody help me to solve this. I would really appreciate. In a certain country, $35$ percent of people ...
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0answers
13 views

Picking Marbles and Estimating the Expected Cost

This is a general version of a game I play. Suppose there is a bag of marbles that consists of 4 different marbles : Silver ...
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1answer
58 views

Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
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0answers
35 views

what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
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2answers
68 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
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1answer
33 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
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1answer
13 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
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1answer
15 views

Discrete-time Markov Chains

I am having trouble understanding this proof from Markov Chains by Norris (1997) How do we get the equality $P_j(X_n=j \text{ for infinitely many } n ) =P_j(X_n=j \text { for some } n \ge m+1)$ ?
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1answer
29 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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1answer
15 views

Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
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1answer
26 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
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0answers
29 views

Regression in Statistics [on hold]

A survey was carried out by a lecturer on a small random sample of her students, in which she asked them individually how much time, $y$, they spent studying and how much time, $x$, they watched ...
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0answers
23 views

the joint probability density

The joint probability density f(x, y) of a pair of random variables X and Y is zero everywhere except on and inside the L- shaped region shown in the following plot, in which the density is some ...
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1answer
32 views

Probability, step function [on hold]

A one-dimensional random walk starts at the origin x = 0 at time t = 0, and takes a random step of either +1 or -1, every second. It takes a total of 4 random steps altogether, with the final location ...
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0answers
24 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
2
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1answer
38 views

How many 3 letter words can you form from 'EEAAP' [duplicate]

How many 3 letter words can you form from 'EEAAP' I think the answer is ${3\choose 3} * 3! + {2\choose 1} * 3 + {2\choose 1} *3=18$. Is this right? ${3\choose 3} * 3!$ = You pick all ...
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2answers
48 views

Runs of white balls in sampling without replacement

There are $m$ white balls and $n$ black balls in a box. Balls are randomly drawn from the box with no return. Denote $X_1$ : number of white balls that been drawn before the first black. For $2 \leq i ...
3
votes
1answer
49 views

How many ways can you choose team of 5 people out of 7 men and 6 women in which there are at least 3 men?

I am confused by this question. I solved it by selecting 3 men first out of 7 men and then selecting 2 people out of 10 remaining person ( 4 men and 6 women ) . So my answer is C(7,3) * C(10,2) = ...
3
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1answer
32 views

Probability of the card following first ace being ace of spades or two of clubs

I am learning probability from Scheldon Ross' book. The question reads like this: A deck of 52 playing cards is shuffled, and the cards are turned up one at a time until the first ace appears. Is ...
1
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1answer
39 views

How do I go about solving this probability question?

I was playing around with dice this morning and flipping them, when a problem suddenly hit me. If I roll $n$ normal six-sided dice, and flip every single dice, what is the probability that the ...
0
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2answers
36 views

Probability and Combinatorics

I am trying to solve example 4.15 here but think the total number of outcomes in the solution is incorrect. This is my reasoning. We have 3 that qualify as best three, say BBB, and 2 as bad say OO. ...
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0answers
16 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 ...