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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2
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1answer
39 views

Is my answer correct? (Devious auction game)

(Taken from here) The question was A man is auctioning a real $20\$$ bill. There are a vast number of bidders. A person may make as many bids as he wants. The starting bid is $5\$$. No $2$ ...
2
votes
1answer
42 views

Find the probability of the product of two random variables

Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $[0,2]$. I am trying to find ${\bf P}(XY\geq 1)$. $${\bf P}(XY\geq 1) = \int_{x}f_X(x)P(Y\geq ...
1
vote
0answers
15 views

Chi-Square Computations

Suppose we have $Y_1,Y_2,....Y_5$ i.i.d. $N(\mu,\sigma^2)$. Find the probability that $S^2/\sigma^2$ is between 0.20775 and 3.2075 where $S^2$ is the sample variance. $P(4*0.20775 < ...
2
votes
1answer
26 views

More than 4 at Even throw of Fair Dice

A dice is being thrown till we get number greater than 4 at Even Throw. What is the Probability of this Event? I have two approaches: Method $1$. The probability of number greater than $4$ is ...
0
votes
2answers
19 views

What is a decision threshold and how does it apply to a statistical power?

I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R. I have a homework problem that says: From the ...
0
votes
1answer
28 views

Why is the Venn Diagram Considered a Special Case of the Formula for Unions

$P( A \cup B) = P(A) + P(B) - P(A \cap B)$ Or in Boolean terms: $P(A + B|C) = P(A|C) + P(B|C) - P(AB|C)$ I've read a lot of information but still can't piece together why this would be considered a ...
0
votes
0answers
32 views

How many ways can I pick 3 marbles from this bag? [closed]

Let's say you have a bag of 300 marbles (100 blue, 100 red, 100 yellow), and you draw 3 marbles from the bag. How many different outcomes are there? (How many different groups of 3 marbles?) Also, ...
1
vote
0answers
26 views

Conditionals of signed measures

My question pertains the definition of regular conditional measures of signed measures defined on product spaces. Consider a Suslin measurable space $\mathcal A=X\times Y$ with the Borel ...
0
votes
1answer
30 views

probability of couples. [closed]

There are 120 people, 10 of them marked BB, 60 of them marked Bb, and 50 of them marked bb. Out of these 120 people, we want to make 60 couples. If picked at random, what percentage of these couples ...
1
vote
1answer
29 views

probability of sequence of exactly 3 heads

A fair coin is tossed 5 times. What is the probability of getting a sequence of exactly 3 heads? Attempt: Sample space = $2^5 = 32$ \number of possible positions of first head = $5 - 3 + 1 = 3$ ...
0
votes
1answer
19 views

Problem involving normal distribution and conditional probability

Here's the problem statement... A sample consisting of 48% men and 52% women is such that the height of men follows a normal distribution with mean 182cm and variance 7cm, and the height of women ...
1
vote
1answer
19 views

Graph Theory: Conditional Expected Value of Product of two Random Variables

Consider a graph with $n$ vertices, where each edge between any two vertices is independently drawn with probability $p$. Let $D_i$ be the degree of vertex $i$. What is $E[D_i \cdot D_j]$? Here is ...
3
votes
1answer
43 views

Class of graphs with symmetric random walk

Let $(V,E)$ be a graph and let $X_n$ be a random walk on the graph. At every step, the walker at $x$ jumps to one of the neighbors drawn uniformly at random among all the vertices $y$ such that there ...
-1
votes
1answer
38 views

Tennis probability question [closed]

Justin plays $3$ tennis matches. The probability that he will win a match is $\frac{3}{4}$. Draw a tree diagram to represent this scenario and provide the sample space. This is from chapter 12 of my ...
0
votes
0answers
20 views

Find $v_k$ the probability of absorption at $N$ if the walk starts at $S_0=k$ for $0 \leq k \leq N$

Supose that $(S_n)_{n\geq0}$ is a random walk on $\{0,1,2,\dots,N\}$ with up prbability of $p$ and down probability of $(p-1)$. Find $v_k$ the probability of absorption at $N$ if the walk starts at ...
0
votes
1answer
26 views

Probability of winning a simple game

Consider two players, A and B start with 8 and 6 stones respectively. A rolls a six-sided die to determine how many stones to take from B. B performs the same task to determine how many stones to ...
4
votes
0answers
35 views

Computing MMSE and conditional expectation

Suppose we have three independent, zero mean, finite variance random variables $V,W,Z$ and where $W,Z$ are Gaussian random variables. These random variables form a new random variable $Y$ ...
0
votes
2answers
24 views

Probability of a dice game: win, lose or draw.

$C$ and $J$ play a game. $C$ always starts. $C$ rolls a fair dice first and wins if he throws an even number. If not, then $J$ rolls the dice. If she rolls an odd number she wins. if neither win it's ...
0
votes
0answers
15 views

Finding probability of being in a certain state in a CTMC.

There are two transatlantic cables each of which can handle one telegraph message at a time. The time to breakdown for each has the same exponential distribution with parameter λ. The time to repair ...
0
votes
1answer
36 views

Shannon's definition of ergodicity

In A Mathematical Theory of Communication (1948) Shannon gives a definition of ergodicity for a Markov process. In order to be ergodic the directed graph of the process must have the following ...
2
votes
1answer
29 views

How to represent $Prob(X_1+X_2 \leq a, X_2+X_3 \leq b, X_3 +X_4 > c)$ with mutually independent random variables?

There are four mutually independent random variables: $$X_i : \Omega \to \mathbb R$$ for $i= 1,2,3,4$ The cumulative distribution function of them is given as $F_i(x_i)$. How to represent ...
19
votes
5answers
308 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real ...
0
votes
1answer
26 views

The difference between a matrix valued random variable and an $n \times p$ matrix of data

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some ...
2
votes
0answers
16 views

Finding the Value of Ties

THE CHALLENGE If "Winning all the time = 1" "Losing all the time = -1" "Tie all the time = 0" What value would "Winning HALF the time" be? Is there an error in my original variables that would help ...
1
vote
2answers
30 views

Binomial probability for sporting event where the order of the first 3 outcomes does not matter

There is a sporting event where team A has $1/3$ chance of winning and team B has a $2/3$ chance of winning. In order to win a team needs to be the first to win 4 matches. There are no ties. The ...
1
vote
2answers
44 views

Changing results of a random process

A very sexist population prefers boys to girls. All parents try various strategies (but not neglect, abuse, or selective abortion) to increase the number of boys, such as giving birth repeatedly ...
2
votes
1answer
40 views

choosing the right value to calculate a probability

A lot of $n$ itms contains $k$ defectives, and $m$ are selected randomly and inspected. How should the value of $m$ be chosen so that the probability that at least one defective item turns up is 0.90? ...
0
votes
1answer
20 views

Expected Value: how to understand this expression?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. ...
0
votes
1answer
31 views

Probability of points in a triangle

$O(2,3)$, $A(2,0)$, $B\left(1,\dfrac{1}{\sqrt{3}}\right)$ are the vertices of $\Delta{OAB}$ on the $\text{x-y}$ plane. Let $\text{R}$ be the region consisting of all points $P$ inside the triangle, ...
0
votes
2answers
21 views

Conditional expectation of continuous RV

Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$? From definition we have: $$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$ Is ...
0
votes
1answer
26 views

Probability of different coloured counters being removed from a bag in a specific sequence

A paper bag contains n counters of each of 4 different colours (i.e. 4 n counters in total). These are selected at random one by one from the bag until they have all been removed. What is the ...
1
vote
3answers
48 views

how to proof formula for general addition rule of three events

can somebody please help to prove the formula for general additional rule of three events? $$P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)$$
0
votes
1answer
30 views

Chebiyshev Inequality

In proving the Chebyshev inequality in Probability theory an important step is to observe that: $P((|x-E(x)|≥a))=P(|x-E(x)|^2≥a^2)$. It is assumed that X has a moment of order 2. Can somebody help ...
0
votes
0answers
22 views

Fewest questions to reach an arbitrary confidence level about the number of words a person knows in a foreign language

I have a list of 13060 Chinese characters that are pre-sorted from most frequently used to least frequently used. I would like to design a simple test to determine the total number (or approximate ...
0
votes
1answer
44 views

Calculate Probability of Winning Draw

If there is a draw each month with 3 prizes, a entrant can only win one of the prizes each month but all entrants will be in the next months draw. There are 120 entrants. What is the probability of ...
0
votes
1answer
32 views

Probability of seeing a patient

Question: A physical therapist has 20 patients. The therapist sees 5 of them once a week, 9 of them twice a week, and the other patients three times a week. If one of the physical therapist's ...
0
votes
0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
2
votes
0answers
39 views

A(nother) variation of the coupon collector's problem

I have come across variation of the coupon collector's problem that goes like this. The coupons are of $n$ different types and in infinite number (or sampled with replacement after each draw, where ...
2
votes
1answer
40 views

student's $t$-distribution

Random sample of $457$ Sample mean = $3.59$ Sample standard deviation $1.045$ Confidence interval from $3.49$ to $3.69$ What is the confidence level? How can I get the answer when sample size is ...
1
vote
0answers
24 views

One dimensional Lazy random walk, $O(1/\sqrt{n})$?

Suppose that we have a Lazy 1-dimensional random walk $X_n$ valued in $\mathbb{Z}$, i.e. $$X_n = \sum_{i}^{n} \xi_i\;\;\;\;\;\;\;\;(\xi_i\;\text{iid}) $$ and $$\frac{1}{4}=P(\xi_1= 1)=P(\xi_1 =-1) ...
0
votes
1answer
51 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
0
votes
3answers
26 views

Given the joint density function for X~Unf(0,2) & Y~Unf(0,3) find Pr(XY < 1)

I have two independent random variables, X~Unf(0,2) & Y~Unf(0,3). Their joint density function is f(x,y) = 1/6 if 0<=x<=2 and 0<=y<=3 else f(x,y) = 0. I'm supposed to find Pr(XY < ...
1
vote
1answer
24 views

Poisson Distribution Problem

The random variable $X$ has a Poisson distribution with mean $\mu$. Show that $$P(X \equiv 1 \bmod 2)=a+be^{c\mu}$$ where $a$, $b$, and $c$ have to be determined. I plugged $X \equiv 1 \bmod 2$ ...
0
votes
0answers
7 views

Ellipsoid confedence intervales?

Are Bonforonni, Scheffe, Multivariate t, and Tukey for simultaneous Confidence intervals are ellipsoid? How can I tell from the form of the interval that it is ellipsoid or rectangular?
0
votes
1answer
42 views

An inequality about the CDF of a random variable [duplicate]

I have to prove that for a random variable $X\sim N(0,1)$, \begin{equation*} \mathbb{P}(X>t) \leq \frac{1}{2}\,e^{-\frac{t^2}2}. \end{equation*} I tried using the fact that for $h(t)=e^{\alpha ...
1
vote
3answers
38 views

What is the probability to get $5$ correct numbers of a $7$-digit number from either the left side or right side?

What is the probability to get $5$ correct numbers of a $7$-digit number from either the left side or right side? For example: The correct number is $1234567$, the number $**34567$ and $12345**$ ...
0
votes
0answers
15 views

Binomial distribution problem with subtractive factor growth

I have a problem that can be modeled using the binomial distribution. I want to know the probability that over n trials I will see k successes with a probability of success $P_s$ and a probability ...
0
votes
1answer
34 views

What is the Difference in the Average and the Mathematical Expectation in the following Problem

Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. The average class size is simply ...
0
votes
1answer
14 views

Find the density of T if $P(T \leq t) = 2 \Phi\left(\frac{x}{\sqrt t}\right)$

Question in the topic. So basically I want to rearrange $\Phi\left(\frac{x}{\sqrt(t)}\right)$ in such a way so I can show that the density of $T$ equals $$(2\pi)^{-1/2}e^{-x^2/(2t)}xt^{-3/2}.$$ $\Phi$ ...
1
vote
2answers
40 views

Piece-wise probability density and cumulative distribution function exercise

Given a random variable $X$ with the density function: $f(x) = a$ if $0 \leq x \leq b$ and $f(x) = b$ if $b < x < a + b$ I want to solve the following exercises regarding this ...