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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Conditional Probabilities

I am faced with this question: 10% of all email you receive is spam. Your spam filter is 90% reliable, that is, 90% of the mails it marks as spam are indeed spam and 90% of spam mails are correctly ...
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1answer
17 views

$MLE$ of $\theta$ when $X_1 , X_2 , …, X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$

How can we find the $MLE$ of $\theta$ when $X_1 , X_2 , ..., X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$ ? $L(\theta) = \prod_{i = 1}^{n} e^{\theta - x_{i}}$ $L(\theta) = ...
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0answers
28 views

Digits of $n$ factorial

With the notable exception of $0$, for large enough $n$, the digits in base $10$ for $n!$ seem pretty much uniformly distributed (I have also checked for other few bases $> 2$). Have anyone ...
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0answers
26 views

probability, birthday paradox: need help to understand the solution

I need help to understand the following solution to a birthday paradox problem. problem:So you have $20$ people. Then let $P=$ # of pairs that share the common birthday. Compute ${\bf E}[P]$, ...
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2answers
20 views

Probability of selecting one of multiple sets of distinct items

Here is the problem I am having: You have a set of items; let's say colored stones. There are 40 stones. 3 Blue, 3 Red, 3 Green, 3 White, 3 Yellow, 3 Purple, 3 Orange, 1 Black, 18 Grey. Without ...
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0answers
10 views

Simple random walk problem, time homogeneity

Similar to a question I've asked before, consider a simple symmetric random walk, $\left\{X_n\right\}$, where the following hold; $$X_n = X_{n-1} + Y_n$$ for $n = 1, 2, 3,\ldots$ $$Y_1, Y_2, Y_3, ...
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3answers
21 views

Probability of drawing an ace on or before ninth card

What is the probability of drawing an ace on or before the ninth card drawn? I could grind out the following computation $$\frac{4}{52}\; +\; \frac{\left( 48\cdot4 \right)}{\left( 52\cdot51 ...
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1answer
25 views

Expected outcome negative for all players?

I'm trying to calculate the expected outcome of a dice game. Without going into the rules and probabilities as it's quite complicated, I was wondering if you could tell me if my results make any ...
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1answer
17 views

If $X$ and $Y$ are random variables, does $\mathbb{E}[{X+Y|X}] = X + \mathbb{E}[Y]$?

If $X$ and $Y$ are random variables, does $\mathbb{E}[{X+Y|X}] = X + \mathbb{E}[Y]$? Note: The answer is still a random variable.$\\$ I should add that I derived this law from the following ...
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0answers
10 views

Combinatorics, expected value, drawing balls from a bag, and customer support

It's been a few years since I've done my CS combinatorics stuff so I'm having a major brain fart here. You put n red balls into a bag. Every t hours you select (n/100) balls from the bag. If a ...
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1answer
19 views

Expected value for the probability function $p * (1 - p)^{n - 1}$

$p * (1 - p)^{n - 1}$ is the function I came up with for this problem: ...
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0answers
10 views

How large a sample will need to be taken to meet the administrator's needs?

A hospital administrator wants to measure average time per hip replacement procedure performed in the institution's operating suite. He is willing to be within 10 minutes of the true value with a ...
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0answers
12 views

If I have a random variable X with a given Probability Density Function, How do I find the PDF of the area of a circle with radius X?

To find the PDF of the area of the circle, do I just substitute the PDF of the random variable X in for the radius in the circle area equation?
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1answer
42 views

Prove $p \times(1 - p)^n$ is a probability function

$p \times (1 - p)^n$ where $p$ is the probability an event will happen after $n$ number of failure attempts. I know I have to show that the sum of the probabilities of all possible events equals one. ...
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0answers
26 views

Continuous Joint Probability Density and proportions

A child prepares a mixture of peanut butter, bananas, and jam for his school lunch daily. Let X and Y denote the proportions of peanut butter and bananas, respectively. The mixture proportions vary ...
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0answers
16 views

Simple Conditional Expectation Problem

$$Y := \begin{cases} \begin{matrix} 5 & \text{w.p. } \frac{1}{5} \\ 2 & \text{w.p. } \frac{1}{2} \\ 1 & \text{w.p. } \frac{3}{10} \end{matrix} \end{cases}$$ Note: $\text{w.p.}$ stands for ...
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2answers
23 views

Expected value for games where you can replay?

Lets just say there's a game where you roll one fair die. If you roll a 1 or 2, you pay 1. If you roll a 3 or 4, you win 2. If you roll a 5 or 6 you roll again until you get a 1, 2, 3, or 4. How ...
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1answer
15 views

probability on constucting contingency table [on hold]

The probability that a married man watches a certain television show is 0.4 and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that ...
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1answer
16 views

Calculating confidence interval - formula

I have the following problem that I get the feeling I'm mixing formulas. ...
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0answers
11 views

combining continuous distributions

If we have several continuous distributions, for example ten Beta distributions, how we can combine them by the linear and log-linear opinion pool methods? I know how to combine discrete ...
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2answers
19 views

A spinner has the numbers 1 to 4 on it.

The probability of spinning a number 4 is 0.1 The probability of spinning a number 1 is 0.6 The probability of spinning a number 2 is the same as the probability of spinning a number 3. Calculate ...
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1answer
19 views

Poisson distribution. $y = 5 (\epsilon + 1)^{-1}$ probability that $y$ is integer

I have ran into trouble that I have no idea how to solve. So the problem is: We have random variable $\epsilon$ which is distributed by Poisson distribution with parameter $\lambda$ = 0.46. Then $$y ...
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0answers
13 views

stationary probability distribution in markov chains [on hold]

What will be the stationary probability distribution of an absorbing state in a given markov chain?
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1answer
7 views

class conditionals and priors

Let's say we have an event A that indicates whether an image contains a person or not. A = 1 indicates that the the image contains a person and z = 0 means that it does not. Assume that L which can ...
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0answers
12 views

Integral with truncated normal distribution

I am attempting to determine closed form equations for several integrals. Suppose $X=N(\mu,\sigma)$ is normally distributed with PDF $f(x)$ and CDF $F(x)$. $$\int_{T}^{\infty} xf(x)dx $$ ...
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1answer
18 views

Probability problem regarding rooks on a chessboard

Eight rooks are placed in distinct squares of an 8 x 8 chessboard, with all possible replacements being equally likely. Find the probability that all the rooks are safe from one another.
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0answers
51 views

Game Theory Help

My professor posted this question and arrived with the following answer, but I am unsure how. Can someone please help me understands this? Alice and Bob play a game Simultaneously Alice picks ...
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0answers
9 views

Proving a disequality involving hypergeometric distribution

How would you prove symbolically the following property? $$H(m_a+1,p_a+1,m,p) < H(m_a,p_a,m,p)$$ where $H(m_a,p_a,m,p)$ is the probability of drawing $m_a$ white balls in a series of $m$ ...
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0answers
14 views

Analysis of Steady State Probability for Markov Process

I have a balance equation, representing a Markov Chain, which yields $$ (K - z) \pi(Z_c = z) = (\lambda_c + (z+1)x) \pi(Z_c = z+1) $$ where K is the maximum state of the server. The term $\lambda_c$ ...
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votes
1answer
37 views

Let $Z=\min(X,Y)$. Find $F_Z(z)$ in terms of $F_X(x)$ and $F_Y(y)$

I think I have the answer to this, I just used independence, but could someone answer to make sure Let $Z=\min(X,Y)$ Find $F_Z(z)$ in terms of $F_X(x)$ and $F_Y(y)$ My basic work goes as follows: ...
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1answer
61 views

Card probability

There are two 10-card decks, consisting of 5 red cards and 5 blue cards each. Both are shuffled separately. One card is then dealt from each deck and compared. This is repeated for all 10 pairs of ...
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2answers
32 views

Probability of tossing regular coins 4 times in a row

Presume we have 3 coins out of which two are regular and the third one has both tails. If we toss a random coin 4 times ( a coin is returned to the group after every toss ), calculate the probability ...
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1answer
32 views

How to find E(XY) and g(x)&h(y) from f(x,y)? [on hold]

When $0\le y \le x \le 1$ and $f(x,y) = 8xy$, how can I find $E(XY)$ and marginal functions $g(x)$ and $h(y)$?
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0answers
9 views

How to show difference amount from scattering.

In our discussion, we need to show that the amount of photons at one wavelength is less than photons at a nearby wavelength. Photons (with a nearby wavelength) are created on the sun's photosphere ...
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2answers
42 views

Hazard function of $\min(X_1, X_2)$

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
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0answers
24 views

Probability bound using Markov and Chebyshev's Inequalities for a 400 coin flip?

Let random variable X = number of heads. Find expectation and variance of X if you bound the probability X >= E(X) + 30 using Markov and Chebyshev's. So if X~Bin(1/2) E(X) = np = 400(1/2) = 200 and ...
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1answer
18 views

Combinatorics Drugs Distribution

Someone already asked this question but I wanted to know why the answer isn't $ {50\choose20} + {30 \choose 20 }+ {10 \choose 10 } $ instead it's $ {50\choose20} \cdot {30 \choose 20 } \cdot {10 ...
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1answer
22 views

Probability Discrete Math Bag

A bag contains 15 different objects, 5 red object, 5 blue objects, and 5 white objects. If 3 objects are chosen at random a) What is the probability that 3 objects of the same color are chosen? b) ...
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2answers
29 views
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2answers
20 views

Given two events, A and B, what is the probability that at most one will occur?

I know this is a very basic question, but my mathematics is coming out a bit wonky. Assume Events A, B are independent. Define: $Pr(A) = 1/6$ $Pr(B) = 1/4$ Let Event C = "at most one event out of ...
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0answers
23 views

Probability theory and counting

(a) $10$ people are being considered to serve as a representative in a committee of $3$ people. Each candidate is equally likely. The President has expressed his support for $2$ of the candidates. ...
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1answer
13 views

Alternative Hypothesis

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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1answer
9 views

Self-Selection Bias Intrinsic to Survey Samples?

I'm new to stats so bear with me in asking this question. I'm sure my novice will shine through. I've noticed that with any survey there is an intrinsic opportunity for self-selection bias (There is ...
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1answer
17 views

Null Hypothesis

Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of 80 dollars. New cost-cutting measures were started and ...
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1answer
16 views

Probability of getting a working product from three groups of products…

The first group of products has one third of defect products and two thirds of working products. The rest two groups have all working products. Find the probability that the random product is working ...
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1answer
13 views

Bigger probability of getting a first class product among two companies…

So, the task is: There are two companies, A and B. The first company A uses the technology which consists of two operations in a row to get a product. The probability of getting a defect product from ...
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0answers
18 views

Bingo Card, probability of hitting multi pattern with a 5-5-4-5-5 draw

There are 15 "B" balls, 15 "I" balls, 15 "N" balls, 15 "O" balls and 15 "G" balls for a total of 75 balls. Numbers are drawn until 5 "B" balls, 5 "I" balls, 4 "N" balls, 5 "G" balls and 5 "O" balls ...
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1answer
48 views

Counting and probability theory problems

(a) A professor designed his final exam as follows: There will be three sections in the exam. Each section has five questions. Students have to pick any two sections to answer, in any order. Within ...
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1answer
17 views

Deciding if this game if fair to a player or not-Probability

For this question what do they mean by is the game fair to the player? Does this mean that the chance of him winning is 1/2 and so chance of losing is also half? For this question I did : Let player ...
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0answers
14 views

Probability (poisson random variable)

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter ...