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
22 views

Find the probability of $P(B|-A)$ given $P(B)$, $P(A|B)$, $P(A|-B)$?

$A,B,C$ not independent I've found $P(-A)$, $P(-B \cap A)$, $P(A \cap B)$ $P(A)$ = $P(-B \cap A)$ + $P(A \cap B)$ Bayes Rule isn't helping me find $P(-A \cap B)$ or $P(B|-A)$ as far as I can see. ...
-1
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1answer
69 views

Three Bags Marble Probability

I found this problem in a maths test, and although I am sure there is a method to solve it, I don't know how. I have three bags. Two bags have identical contents- 1 black marble and 2 white ones. The ...
0
votes
1answer
47 views

Expectation of a maximum function

Given for some $y_k$, \begin{align} w_k &= \begin{cases} 0 &\text{w.p. } \frac{1}{4} \\ 1 &\text{w.p. } \frac{2}{4} \\ 2 &\text{w.p. } \frac{1}{4} \end{cases} \\ L_k(y_k) &= ...
1
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3answers
48 views

Distribution of the difference of two uniformly distributed variables subject to a condition!

Let $A$, $B$ and $C$ be three independent uniformly distributed random variables on $(0,1)$. The variables are admissible if and only if $a<b<c$. I want to find the distribution of $B-C$ given ...
2
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2answers
108 views

a problem for the probability of distributed objects

17 books and 17 journals are randomly distributed among 17 boy and 17 girl students such that each student gets one item. Find the probability that at least one boy gets a journal.
2
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1answer
45 views

Conditional probability times conditional probability

I am not sure if the Bayes rule can be used under the following conditions, can anyone help me to determine if the equation is right? $$ P(A\mid B) \cdot P(B\mid C) = P(A,B\mid C) $$ $$ P(y\mid A, B) ...
1
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0answers
80 views

MLRP of random variables and order statistics

Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy MLRP. (ie $ \forall a<b$ $ ...
2
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2answers
756 views

Rolling two dice 10 times, what is the probability of getting all possible “doubles”?

Rolling two dice 10 times, what is the probability of getting all possibble "doubles": (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) among our rolls? For instance, ...
0
votes
1answer
535 views

Solving problem with negative tcalc value. Please help.

Using the above data check to see if the average cholesterol determination was statistically the same or not for two labs. alpha= 0.05. Use a independent design. My problem is that I got ...
1
vote
1answer
85 views

Probability of a 9 shrews to have a total mass of 100g. Please help!

The mass of a species of shrew is approximately normally distributed with a mean of 10 grams and STD of 5 grams, then: A. What is the probability that a random sample of 9 shrews had a total mass of ...
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6answers
3k views

Why is not the answer to all probability questions 1/2.

Ok, I know this is wrong, but I want someone to tell me why. Let's take a normal heads tails example of a fair coin. The probability of getting head = 1/2. And I write this is because, ...
3
votes
1answer
62 views

probability of a function f(x) to be increasing

Suppose $f(x)=x^3+ ax^2 + bx +c$ . Now a,b,c are chosen respectively by throwing a dice 3 times. Now find the Probability that $f(x)$ is a increasing function ? MY APPROACH : i really have given a ...
0
votes
2answers
48 views

Random var. Y with pdf $f_Y(y) = 4y^3$. Show that $-2\ln (Y^4)$ ~ $X_{(2)}^2$.

Let Y be a random variable which has pdf $$f_Y(y) = \begin{cases}4y^3, & 0 < y < 1, \\ 0, &\text{elsewhere}.\end{cases}$$ Show that $-2 \ln (Y^4)$ ~ $X_{(2)}^2$. Could anyone get me ...
2
votes
1answer
45 views

Prove that $P[X>\epsilon] \leq M(t)/e^{\epsilon t}$

Prove that $P[X>\epsilon] \leq \dfrac{M(t)}{e^{\epsilon t}}$ Looks like Markov's inequality, it's very easy to derive for $t>0$ $P[X>\epsilon] =P[Xt>\epsilon t]=[e^{Xt}>e^{\epsilon ...
1
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2answers
72 views

Calculate the probability density function of a segment' length

There is a circle, with radius $R_0$, is centered on the origin. Choose a uniformly random point $(x,y)$ on the circle. Q1: Calculate the PDF $f(x,y)$ Q2: Calculate the PDF $f(l)$ of the length of ...
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votes
1answer
38 views

Quesion on Probability

A machine works with at least 2 batteries.the probability of any 1 battery not working in less than 50 hours is 0.2 . Find the probability that the machine will function satisfactorily for at least 50 ...
0
votes
2answers
940 views

Version 2:Help finding the probability that $Ax^2 + Bx + C$ has real roots?

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $AX^2 + BX + C$ has real roots? I am given a hint that if ...
1
vote
2answers
214 views

Distribution of angle of two dimensional normal vector

The original subject is: Suppose random variables $X$ and $Y$ are independent and both follow the Normal distribution $N(0,\sigma ^2)$. 1) Prove $U=X^2+Y^2$ and $V = \frac{X}{\sqrt{X^2+Y^2}}$ ...
0
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1answer
193 views
1
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0answers
24 views

independent random variables made from chi-square distribution

if $x_1$,...,$x_n$ are r.s from N(0,1), how can we prove $\frac{x_i^2}{\sum_ {j=1}^{n}x_j^2}$ and $\sum_ {j=1}^{n} x_j^2$ are independent?
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1answer
147 views

Dice Math, odds and probability

DISCLAIMER: This will be done with fake money for education purposes. I'm working on a Dice simulation site for a school project, where you play the house and you as a player have to get a higher ...
0
votes
1answer
43 views

Finding P value

I have these observations $(2,3.2,3.8,2.5,3.3,2.8,3.0,3.4)$ from $X \sim N(\mu,\sigma^2)$ and i want to calculate the $P$-value testing $H_0: \mu =3.2$ against $H_1 \neq 3.2$ with $\sigma = 0.6$ ...
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votes
1answer
34 views

Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

Given t statistic $t = \frac{\bar{x} - \mu}{s/\sqrt(n)}$ prove $\bar{x}$ and $s$ are independent...you do not need to prove it...just tell the methods. One approach that I think of is Let $X = ...
3
votes
0answers
124 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distribted over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC ...
3
votes
1answer
383 views

Elementary proof of geometric / negative binomial distribution in birth-death processes

The birth-death process concerns a population of $n_0$ individuals, each of which reproduce and die at a constant rate as time $t$ increases from $t=0$. Each individual splits into two individuals ...
5
votes
1answer
175 views

How many times do you have to toss a coin so that the probability that #heads = #tails is 0.01?

I thought it would be something like: $Binomial(n, \frac{1}{2})$. So $0.01 = \binom{n}{\frac{n}{2}}(\frac{1}{2})^{\frac{n}{2}}(\frac{1}{2})^{\frac{n}{2}}$, because half of $n$ should be heads and the ...
0
votes
1answer
60 views

A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
0
votes
1answer
35 views

Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
0
votes
3answers
63 views

Resolving this probability paradox

We toss a fair coin 100 times. We get 100 heads in a row. Now we toss this coin one more time. Two things, each individually, make sense to me: The probability of heads coming up again is 1/2 ...
0
votes
1answer
27 views

Probability condiioned on two variables

Given random variables $X, Y, Z$, when does $p(X|Y, Z) = p(X|Y)p(X|Z)$? Is such a transformation ever justified?
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1answer
55 views

How to find the proper $\alpha$ to satisfy the 80/20 rule for the Paretor Distribution

Suppose we have the CDF for the Pareto Distribution given by: $$ P(X \leq x) = 1-\left(\frac{x_m}{x}\right)^\alpha \;\;\;\;\;\;\;\;\;\; x \geq x_m$$ What is the intuitive way to find the alpha for ...
2
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1answer
25 views

recalling probability rule

let us consider following picture ,we have given some initial data and also list of questions which we should solve i would like to solve part $e$,as i remember in this case i should use Morgan ...
2
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2answers
584 views

Conditional expectation of independent random variables

Suppose $X, Y, Z$ are random variables and $X, Y$ are independent to each other. Can I write $\mathrm{E}[XY|Z]=\mathrm{E}[X|Z] * \mathrm{E}[Y|Z]$ ?
0
votes
1answer
32 views

random events, show if $P(B')=0.4$ then $P(A\cap B') \le 0.4$

Let $A,B$ be random events included in $\Omega$ space. Show that if $P(B')=0.4$ then $P(A\cap B') \le 0.4$ so my idea is to show that $P(A \cap B') \subseteq P(B')$ then we have $P(A \cap B')\le 0.4$ ...
0
votes
2answers
36 views

When the game is fair

One player throws dice twice. If he has 2 x 6 on the dice he is receving 8*a. If he has one 6 he will collect 4*a. Otherwise (when he has no 6 at all) he is paying a. For which value of a game is ...
2
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0answers
49 views

We are giving $m$ prizes to $n$ people at lottery…

We are giving $m$ prizes to $n$ people at lottery... Question A: What is the probability that no one will get more then one prize (assume that $n\ge m$). Question B: What is the probability the ...
1
vote
2answers
121 views

$R^2$ is uniform on (0,1)

Let (X,Y) be uniformly distributed in a circle of radius 1. Show that if R is the distance from the center of the circle to (X,Y) then $R^2$ is uniform on (0,1). This is question from the Simulation ...
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2answers
252 views

what the central limit theorem says

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
1
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2answers
89 views

tail probabilities for the sum of independent Laplace random variables

How might I find tail probabilities (pr X>x), or a reasonable approximation, for a variable that is the sum of independent Laplace random variables?
0
votes
1answer
71 views

finding joint pdf

Suppose the continuous random variable $Y = X + V$, where $X$ & $V$ are continuous random variables with parameters ($\mu_x,\sigma_x$) & ($\mu_y,\sigma_y$). How should I go about finding ...
0
votes
1answer
39 views

What is the name of this Inequality

I came across the inequality below and I don't know its name or how was it derived, any ideas? Let Po($\lambda$) be a Poisson random variable with density $\lambda$. If $K> e^2 \lambda$ then: ...
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2answers
41 views

message plus response probability problem

Suppose I need to send a message to someone. So that I will know the message was received, they send one back. Suppose there is a 10% chance that the message sent will get lost. Similarly, there is a ...
0
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1answer
127 views

Markov chains example

Your exam could be marked with a range of possible grades, simplified as on the following state diagram: To begin with the chances are that you will pass with a standard result. Each 45 minutes ...
1
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1answer
120 views

Geometric random variable $X$, $Pr(X\ is\ even) =$?

Original Question: Toss an unfair coin until we get HEAD. Suppose the total number of tosses is a random variable $X$, and $Pr(HEAD) = p$. What is the probability that $X$ is even? Denote this event ...
0
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2answers
1k views

expected value of this markov chain

Question: A bag contains 3 white chips and 3 red chips. You repeatedly draw a chip at random from the bag. If it's white, you set it aside; if it's red, you put it back in the bag. After removing all ...
2
votes
1answer
53 views

Coarser cyclic decomposition of Markov chain

For a irreducible Markov chain with period $d$ there is a standard construction which shows that the state space can be partitioned into $d$ sets $C_1, \ldots, C_d$ such that $P(x,y)>0$ only if $x ...
1
vote
1answer
61 views

Help with using the “Inclusion–exclusion principle”

I have question at probability that I need to use the "Inclusion–exclusion principle"... Hand of bridge is 13 cards that picked up randomly. What is the probability that we will have a King and Ace ...
2
votes
3answers
38 views

Deriving a property regarding variance.

I am studying for the P-exam for actuaries and I've encountered a property that said, $Var(x\pm y)=Var(x)+Var(y)$ I come from a math major and it has been years since I was taught statistics or ...
2
votes
1answer
132 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$\mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{\alpha W_t} dt}{\int_0^T e^{-\alpha W_t} dt + \int_0^T e^{\alpha W_t} dt} \right]$$ For $\alpha \in ...
2
votes
1answer
53 views

Simplification of combinatorial formula

I encountered the following formula while working out a problem in a different context and could not figure out a way to simplify it. After spending a fair amount of time on it and not really making ...