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
17 views

Simplex Algorithm (Exercise 3.11.33 in Grimmett and Stirzaker's Probability and Random Processes)

There are $n \choose m$ points ranked in order of merit with no matches. You seek to reach the best, $B$. If you are at the $j$th best, you step to any one of the $j - 1$ better points, with equal ...
0
votes
1answer
18 views

estimating probabilty [on hold]

I would like some help with a probability problem. Given that $23.5$% are obese, $22.7$% of Americans are smokers, and $4.7$% are both obese and smokers. Estimate the probablity of 1) A person that ...
-1
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0answers
16 views

Confusing Conditional Probability question 68 [on hold]

The four top tennis players in the world A, B, C, and D are invited to a special tournament where the winner gets one million dollars. In round one, Player A plays player D and player B plays player ...
5
votes
0answers
29 views

markov chain: 2 state chain

I have a machine. It has two states, broken or working. If it is working, then it will be broken with probability $q=0.1$. If the machine is working, I will make \$1000 dollar a day. If it is broken, ...
-1
votes
1answer
22 views

Waiting time for two independent poisson processes

Order of Events in Poisson Processes Assume that you have two independent Poisson process, $N_1(t)$ with rate $\lambda_1$ and $N_2(t)$ with rate $\lambda_2$. The probability that $n$ events occur ...
2
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1answer
35 views

Probability of a random Permutation [on hold]

Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last ...
1
vote
1answer
18 views

Expected random generation

Background: I've been reading about how Dota deals with its random generation. There's another question on Gaming.SE about this, but it doesn't give a formula, which is what I'm looking for. ...
2
votes
1answer
30 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
0
votes
0answers
7 views

Derive distribution of a random variable given an observed perturbation

I have a process by which some initial value $x_0$ is perturbed by $\epsilon$ to $x_{obs}$, where $\epsilon$ is a random number drawn from a PDF $p(\epsilon)$. Given a particular observed value ...
0
votes
1answer
34 views

Poisson probability of an event A before event B

I'm trying to calculate the probability of two poisson processes events happening one before the other, with two different $\lambda$s. The way I see it, I can word it as the probability of event $A$ ...
2
votes
0answers
20 views

NHL Lottery - Figuring out % chance at top 3 pick

The NHL has changed their draft lottery system this year to match the NBA's, sort of. The first 3 picks are up for lottery, with weighted odds based on standings. I run a site that simply just ...
0
votes
2answers
25 views

How to find the limit of a matrix $P^n = UD^nU^{-1}$ where $D$ is a diagonal matrix of eigenvalues and $U$ a matrix of eigenvectors?

If we have a matrix where $P = UDU^{-1}$, where $D$ is a diagonal matrix of real eigenvalues that are less than or equal to 1, and $U$ is the corresponding matrix of eigenvectors, how can we show that ...
1
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2answers
29 views

How do you interpret conditional probability when two events are switched?

Before I pose my question, I want to emphasize that I am not seeking a homework help or steps on how to derive the answer, for I already know the solution, and how to get it. What I am seeking is, how ...
2
votes
1answer
48 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
3
votes
1answer
26 views

Urn probability replacement problem

An urn contains $10$ red and $10$ white balls. They are taken out at random one at a time. Find the probability that the fourth white ball is the fourth, fifth, sixth or seventh ball drawn if the ...
0
votes
1answer
43 views

Expected number of red balls in urn

We toss balls into urns. Denote with $x$ the number of balls in an urn. And $x_r$ denotes the number of red balls. The share of red balls among the balls is denoted as $P$. We toss these balls into ...
1
vote
2answers
29 views

Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...
0
votes
1answer
25 views

Simulate random variable with PDF $x^2+\frac4 3x$ on $[0,1]$

Consider $X$ a random variable with the following density function: $f(x) =$\begin{cases} 0, & \text{x ∉ [0,1]} \\ x^2+\frac4 3*x, & \text{x \in [0,1]} \end{cases} I need to write a ...
-1
votes
5answers
78 views

Deck of Cards Stats Probability Question [on hold]

Randomly select two cards in sequence from a full deck of 52 cards, what i s the probability that the first one is a King given that the second one is a King. If someone can please help me with this ...
0
votes
1answer
22 views

Finding $P(C)$ with Bayes's Theorem

We have two events $C$ and $D$ such that $0<P(D)<1$ and a $P(C|D)=P(C|D^{c}) = \frac{1}{3}$. I am wondering if it is possible to calculate $P(C)$ from only this information. I've tried using ...
1
vote
1answer
36 views

Expected value of a poisson process

I've been searching for a while but I can't seem to figure out how to find the expected value of a poisson process up to an arbitrary time. Let {$N(t),t≥0$} be a Poisson process with rate $λ$. How ...
1
vote
2answers
18 views

Probability of an event based on percentage in fixed lapse of time.

I am a software engineer. I am also a former triathlete that rides with a large group of friends every time we have a chance. i am trying to come up with a little software to distribute among us ...
0
votes
0answers
10 views

Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
0
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0answers
32 views

Probability - coin flip with known tails count and probability of heads is known.

First coin toss out of $10$ will always be tails. Five tails will appear in total (including the first tail) out of $10$ flips. The probability of heads is $0.3$. What is the probability for exactly ...
2
votes
1answer
54 views

Number of inversions

Compute the sum of the number of inversions that appear in the elements of $S_n$. In other words find the total number of inversions that the elements of $S_n$ have combined. I mean how can we ...
0
votes
0answers
20 views

Average number of $5$-card draws before all $52$ cards in a deck are drawn.

So an interesting question was brought to me today and I'm not sure how to formulate the equation to answer it. A person draws $5$ cards from a deck, writes the cards down, puts the cards back in the ...
0
votes
1answer
28 views

Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
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0answers
17 views

Normal Approximation to the Binomial

I need to solve this problem using Normal Approximation to the Binomial Distribution to check if the value is similar to the one that I found using the Binomial distribution. Question: What is the ...
0
votes
1answer
33 views

calculating probability using mean and standard deviation [on hold]

The time that it takes to assemble a piece of machinery is well modeled by the normal distribution with mean of 72.9 minutes and standard deviation of 8.55 minutes. What is the probability that it ...
0
votes
1answer
28 views

Binomial distribution births

I am trying to solve this problem using binomial distribution. What is the probability that in a group of $100$ people, $10$ of them were born in either March and April. Binomial distribution ...
1
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2answers
33 views

Probability of the next number in a random sequence being the largest seen so far

Suppose I have a uniform random number generator producing a sequence of random numbers in the range $0...100$. I am trying to work out what the probability is that the $n^{th}$ number in this ...
-2
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0answers
19 views

Probability of independent event, Slotting machine model [on hold]

A slot machine has four separate wheels that rotate independently. On each wheel are four pictures of a lemon and one picture of a cherry. Each time the slot machine level is pulled, one picture on ...
2
votes
2answers
28 views

What is the probability that nobody receives the same ranking twice?

Four players compete in a tournament and are ranked $1$ to $4$. They then compete in another tournament and are again ranked from $1$ to $4$. Suppose that their performances in the second ...
0
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0answers
28 views

How to solve the Probability Markov chain system of equations

I have this system of equations from a 2-D Markov chain (see the figure. How can i calculate the coefficient matrix, state probability vector and the constant vector from this system of equations. ...
-2
votes
2answers
26 views

If $\mathbf{E}(e|x) = 0$, then $\mathbf{E}(h(x)e) = 0$ for any function $h(x)$ [on hold]

Consider a random scalar $e$ and a random vector $x$. Let $\mathbf{E}(e|x) = 0$. Show that $\mathbf{E}(h(x)e) = 0$ for any function $h(x)$ I am asked to show this, but I have no clue where to ...
0
votes
0answers
15 views

Moments of censored exponential distribution

I have a question as to whether my calculation of moments of censored exponential distribution is correct. I have two random variables $T_A=\min(\tau,t_1)$ and $T_B=\min(\tau,t_2)$, where $t_1<t_2$ ...
0
votes
2answers
28 views

The skewness coefficient for given pdf?

$$f(x)= \begin{cases} 0.5-\frac{x}{8},& 0\le x \le 4 \\ 0, & \text{otherwise} \end{cases} $$ I have found $E(X) = \frac{4}{3}$ and $Var(X) = \frac{8}{9}$ Th problem says that the skewness ...
1
vote
1answer
20 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
-4
votes
0answers
30 views

statistical probablity [on hold]

A man tosses two fair dice. What is the conditional probability that the sum of the two dice will be 7, give that (i) The sum is odd, (ii) the sum is greater than 6, (iii) the two dice had the same ...
3
votes
2answers
32 views

Pairwise independence vs independence

Two fair dice are thrown. We have three events: A: The first die shows an odd number B: The second die shows an even number C: Both are odd or both are ven Show that $A,B,C$ are ...
2
votes
1answer
29 views

Finding cdf, percentile, variance, and standard deviation from pdf.

$$f(x) = \begin{cases} 2(1-\frac{1}{x^2}) & \text{if }1\le x\le2 \\ 0 & \text{otherwise} \end{cases} $$ Compute the CDF of X: $$ \int^X_12(1-\frac{1}{y^2})dy = 2x+\frac{2}{X}-4 $$ So I ...
0
votes
1answer
18 views

Ehrenfest chain

In the Ehrenfest model, let $X_n$ denotes the number of balls in the left urn. And there are $N$ balls total. When we calculate $P(X_{n+1}=i+1|X_n=i, X_{n-1}=i_{n-1},...,X_0=i_0)$, why don't we take ...
0
votes
0answers
10 views

Finding recurrent states given a Markov chain

I have trouble in approaching the problem where: Consider a Markov chain $X_n$ , $n ≥ 0$, with state space $S = N = {0, 1, ...}$ and transition function $$ p(x,y) = 1/7, y=0 $$ $$ p(x,y) = 2/7, y∈ ...
1
vote
1answer
37 views

I roll a fair die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears.

I roll fair a die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears. Not sure how to go about this.
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0answers
15 views

Poisson Counting Insurancee example [on hold]

An insurance company finds that for a certain group of insured driver , the number of accidents over each 24 hours period rises from midnight to noon and then declines until the following ...
2
votes
3answers
26 views

Probability of an even number of sixes

We throw a fair die $n$ times, show that the probability that there are an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$. For the purpose of this question, 0 is even. I tried doing ...
-1
votes
2answers
39 views

What is the probability that when a deck of cards is shuffled and dealt, exactly 3 of the 4 aces will be dealt within the last 20 cards? [on hold]

I am trying to figure out this problem, I think that it is a "permutations with repetition" type of question.
3
votes
1answer
25 views

Probability: Finding the Number of Pears Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
1
vote
1answer
15 views

Probability density function in Rayleigh distribution

It says that $$ f(x;\theta) = (x/\theta)e^{-x^2/(2\theta^2)}, x>0 $$ is the Rayleigh distribution. And asks to verify that $f(x;\theta)$ is a legitimate pdf. Can you explain how to verify ...
0
votes
1answer
31 views

Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...