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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Probability - conditional

The probability that bulbs are detected faulty if they are defective is 0.95 and the probability that bulbs are declared fine if in fact they are fine is 0.97. If 0.05 of the bulbs are faulty, what is ...
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0answers
9 views

How to prove that the set of all exchangeable events is a sigma-algebra?

Let $ {X_n}_n $ be sequence of identical R.Vs Mark by S the set of all sequences available from it. An exchangeable event is $E\subset S $ which is not sensitive for finite permutations. ...
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0answers
17 views

The probability that two matrix vector products are equal

Consider a random $n$ by $n$ circulant matrix $M$ whose first row entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first ...
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0answers
17 views

Probability -dividing into groups

In how many ways can 12 people be separated into 3 groups of 4 if the 12 comprises 6 pairs of partners? We must keep partners in the same group, but we do not distinguish between the group $(a, b, c, ...
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1answer
17 views

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart [on hold]

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart??
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2answers
34 views

dice probability - same 2 dice in 6 dice rolls

I have this simple probability problem that I am not sure I solved correctly. I am not interested in formulas, but rather the thought process of how to solve it. Suppose we roll six 6-sided dice that ...
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3answers
35 views

Random variable with 2 distribution functions

Just a question here, Given a random variable $X$ defined in a probability space, is it possible to have more than one distribution function $F$ ?
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1answer
29 views

Probability that a year contains 53 Mondays

The question: Find a probability that a year chosen at random has 53 Mondays. Now I know in a non-leap year, probability of getting 53 Mondays is $\frac{1}{7}$ and in a leap year, probability of ...
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1answer
30 views

Drunk Passenger Probability question [duplicate]

I don't know how to solve this question. A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. For convenience, lets say that ...
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1answer
13 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
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3answers
19 views

The dice is rolled 10 times and the results are added with given conditions.

Q: A dice has one of the first 6 prime number on each its six sides ,with no two sides having the same number .the dice is rolled 10 times and the results added.the addition is most likely to be ...
2
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1answer
20 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
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3answers
318 views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
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1answer
25 views

Expected number of customers sitting on correct places

In a shop customers are given a seat number before entering the shop in the order 1,2,3,...,n but after entering the shop they sit in a random order not related to their seat number. what is the ...
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2answers
16 views

how Find the probability that the committee will consist of the following all dentists

A committee of four people is to be formed from six doctors and eight dentists. Find the probability that the committee will consist of only dentists
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1answer
23 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
-5
votes
1answer
52 views

Probability of randomly selecting one student from each of three cities [on hold]

The geographical distribution of hometown of some 80 students at DLSU-D is given as: 50 from Cavite, 10 from Laguna, and 20 from Manila. Suppose three students are selected. Find the probability that ...
1
vote
1answer
21 views

Expected value of function of minimum between two random variables

Two independent random variable $X,Y$ are distributed on $[0,\infty)$ according to the cumulative distribution function $F(x)=1-(x+1)^{-2}$. Let $Z=\min(X,Y)$. Determine $E\left[\frac{Z}{Z+2}\right].$ ...
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0answers
19 views

An inequality related to supermartingale?

Let $X_n\ge0,n\ge0$, be a supermartingale. Show that $CP(\sup X_n>C)\le EX_0$. I tried to use the inequality supermartingale satisfies, which is $E(X_n|\cal {F_{n-1}})$$\le X_{n-1}$. However, ...
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0answers
27 views

How can I formed as below permutation problem

Hi I am writing a program and i encouraged the below permutation problem and need your help. There are 4 boxes: 3 of them have 2 balls The one box has 1 balls. The question is what is the ...
0
votes
2answers
26 views

Anagrams contained within random strings

What is the probability that a random string of length $n$ will contain an anagram of a shorter string of length $k$? E.g., you generate a string of 50 random letters, repetitions allowed, what are ...
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0answers
27 views

Probability distributions associated with Markov chain

Let's say I have a Markov chain, with all the transition probabilities known, and there's a cost associated with each transition. The cost for transitioning from node $a$ to node $b$ is given by the ...
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1answer
18 views

5-Card Poker Two-Pair Probability Calculation

Question: What is the probability that 5 cards dealt from a deck of 52 (without replacement) contain exactly two distinct pairs (meaning no full house)? Solution: ...
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1answer
25 views

How to calculate the odds of a 5x5 Bingo game?

I don't have a mathematics background, but am trying to calculate what the theoretical odds of winning a 5x5 bingo game is if 5 numbers are drawn. Eg board: ...
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1answer
25 views

Prove that X has a chi square distribution

If $X_1,\dots ,X_{30}\sim N(1,\sigma^2)$ and $\hat \sigma^2 = \frac{\sum(X_i-1)^2}{30}, $ then show that $30\,\hat σ^2 /σ^2$ has a chi-square distribution with $30$ degrees of freedom.
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0answers
15 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. If this is not acceptable - administrators please delete, ...
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0answers
16 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
0
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1answer
24 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
2
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1answer
22 views

An Example of sequence of R.V with $E(X_n) = X_0$ but $E(X_n^{1/2}) \to 0$

I need an example of $\{X_n\}_n$ be a sequence of nonnegative, random variables, with the same finite expected value $E(X_n)=\mu_0$, that obeys: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
2
votes
1answer
14 views

Expectation of max absolute value of a Gaussian vector

Let $X$ be a joint Gaussian vector of dimension $k$ with zero mean and covariance matrix $K$ (where $K$ may not be diagonal). I am interested in sharp estimates on $$\mathbb{E}\max_{i=1,2,\ldots,k} ...
0
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2answers
30 views

What is the probability that both the numbers are odd with given conditions?

2 Numbers are selected at random from the integers 1 through 9.If the sum is even,find the probability that both the numbers are odd. My approach: A:Event of getting sum as even, B:Event of ...
2
votes
1answer
74 views

Monty Hall problem again (from Grimmet and Stirzaker)

Grimmet and Stirzaker Exercise 1.4.5.2 In a game show you have to choose one of three doors. One conceals a car, 2 conceal goats. You choose a door but the door is not opened immediately. Instead ...
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2answers
28 views

If $x_i$ is from a random sample is $Var(\bar x \mid x_i)=0$?

If $x_i$ is from a random sample, is the conditional variance of the mean (or the sum of squares, really any statistic based on $x$) just treated as a constant? I saw this in a OLS variance of a ...
2
votes
2answers
32 views

Infinite sequence of exponentially distributed random variables

Consider an infinite sequence of exponentially distributed random variables, $X_k$, where$ k \in \{1, \ldots, n\}$ with $\lambda = 1$. I am trying to evaluate: $$\lim_{n\to\infty} \frac{\max_{1 \leq ...
0
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1answer
38 views

Probability mass function (pmf)

Let's say we have an computer program with a loop. Inside the loop the program encounters a problem with probability p, in which case it increases a counter by 1 and prints out a warning. If the ...
0
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2answers
22 views

I need help with finding a probability [on hold]

Here is the question A bag contains 25 coins. There are 6 quarters,4 dimes,5 nickels,and the rest are pennies One coin is drawn at random. Find the probability that it will be a nickel
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0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
0
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1answer
28 views

Question related to Square Integrable Martingale

Let $X_n$ and $Y_n$ be martingales with $EX_n^2<\infty$ and $EY_n^2<\infty$ for all $n$. Show that $EX_nY_n-EX_0Y_0=\sum_{m=1}^nE(X_m-X_{m-1})(Y_m-Y_{m-1})$. I tried to expand the right ...
2
votes
2answers
27 views

Probability that two sets do not intersect

I'm trying to understand this simpler problem so I can apply the process to a more difficult homework problem. Let $U$ be a set with $n$ elements. Select $2$ independent random subsets $A_1, A_2 ...
3
votes
1answer
42 views

Law of Large Numbers - utility/difficulty of various versions.

This may or may not be an answer to Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?, depending on ...
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0answers
26 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
1
vote
1answer
36 views

equation for probability stumper?

Jane says she was in town for two consecutive days last week (7 days), but won't say more. For each given day, what was the probability she was in town that day? I know there are 6 possible ...
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votes
2answers
32 views

I need to find the mean of students who got below the 70th percentile [on hold]

$50$ children took a test. $5$ children got $100\%$, another $10$ got $90\%$, and yet the average class score was $45\%$. What is the mean score for the children below the 70th percentile? a. ...
2
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1answer
22 views

Finding the distribution of the sum of n independent random variables having exponential distributions

My graduate level probability class asks us to calculate the distribution of the sum of n independent exponentially distributed random variables. I am trying to perform many convolutions but it gets ...
1
vote
1answer
37 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
1
vote
2answers
41 views

Help needed to solve probability problem

I am trying to solve the following problem. A fisherman is equally likely to go fishing at one of the three ponds A,B,C. The probability to catch fish if he cast his rod at pond A is 0.4 , at pond B ...
0
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0answers
24 views

In an incomplete market does every payoff function admit at least two arbitrage-free pricings?

Consider an arbitrage-free (not necessarily complete) market. Prove or disprove the following assertion. If the market is incomplete, then every payoff function $A : \Omega \rightarrow \mathbb{R}$ ...
2
votes
1answer
60 views

How to remember these probability results?

If $A,B$ and $C$ are $3$ events, then $P$(Exactly one of $A,B,C$ occurs)$=P(A)+P(B)+P(C)-2[P(A \cap B)+P(B \cap C)+P(A \cap C)]+3P(A \cap B \cap C)$ $P$(Exactly two of $A,B,C$ occur)$=P(A \cap ...
0
votes
1answer
22 views

Basic conditional probability question [on hold]

$\sum_{c}p(a|c)p(c|b)=p(a|b)$. Does this equation hold true? If it is true, how to prove it mathematically?
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4answers
62 views

What is the probability of drawing 3 balls such that none of them is red?

Given a bag containing $8\ \color{red}{red}$ balls and $4\ \color{green}{green}$ balls, what is the probability of drawing $3$ balls at random such that $\mathbf {none}$ of them are ...