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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-1
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2answers
53 views

almost sure convergence given density [closed]

my problem: Let $X_n$ be iid random variables with density $f(x)=\frac{1}{2}x^{-2}1_{\{|x|>1\}}$. Show that $\frac{X_n}{n}$ does NOT converge almost surely. Can anybody help me?
-5
votes
0answers
55 views

5 girls and 3 boys are arranged randomly in a row. Find the probability that…

5 girls and 3 boys are arranged randomly in a row. Find the probability that: a) the 5 girls are next to each other,= 2/28 b) the 3 boys are next to each other,=3/28 c) there is one boy on each end, = ...
0
votes
2answers
23 views

Why isn't the number of rolls bounded from above

I read this question:How to generate a random number between 1 and 10 with a six-sided die? And read in the comments that "since 6 is not divisible by all the factors of 10", there is no method that ...
3
votes
1answer
23 views

A tool weight is distributed normally with mean = $2265.4$. Given that 14% of the tools' weight are above 2278.36. what is the standard deviation?

A tool weight is distributed normally with mean = $2265.4$. Given that 14% of the tools' weight are above 2278.36. what is the standard deviation? Here the solution: denote $X$ as tool's ...
0
votes
1answer
24 views

Empty intersection in chain rule for probability [duplicate]

I am looking at the expansion of the chain rule for probability. $$ P\left(\bigcap_{k=1}^nA_k\right)=\prod_{k=1}^nP\left(A_k\middle|\bigcap_{j=1}^{k-1}A_j \right) $$ if ...
0
votes
1answer
51 views

Sequence of random variables that follow WLLN but not SLLN

I have to contruct a sequence of random variables that follow Weak Law of Large Numbers, but don't follow Strong Law of Large Numbers. Can canyone give me any hint please? Basically i need to choose ...
0
votes
0answers
19 views

Create a recursion here [duplicate]

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain either exactly two adjacent chairs or no adjacent chairs. I had this question before, but I ...
1
vote
1answer
40 views

Probability with exponential random variable

Machine $1$ is currently working. Machine $2$ will be put in use at time $t$ from now. If the lifetime of machine $i$ is exponential with rate $\lambda_i=1,2$, what is the probability that ...
1
vote
1answer
37 views

Unfairish Probability

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability ...
3
votes
3answers
108 views

Card Game Bridge Probability

I'm trying to self-educated myself and I bought a probability book, which has this interesting question. It says not to look at any resources before you try it, but you may use a calculator. In the ...
0
votes
0answers
26 views

Problem with random permutation and conditional probability

Let $\pi_1,...,\pi_n$ be a random permutation of numbers $1,...,n$. If you are told that $\pi_k > \pi_1,...,\pi_k > \pi_{k-1}$, what is the probability that $\pi_k = n$? What I've tried: Let ...
7
votes
1answer
235 views

Magic 8 Ball Problem

This problem is probably simple enough to have an analogous problem, I just don't know the name so I'm going to describe it and hopefully somebody can point me in the right direction. The problem is ...
1
vote
1answer
52 views

Contradiction on a probability computation

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
1
vote
0answers
31 views

Sum of infinite dimensional random variables

If $X_i$ are Independent and Identically Distributed (IID) vector valued variables with positive mean, and finite variance, then with Chebyshev's inequality, we know that their sum $S_n=\sum_{k\leq n} ...
2
votes
0answers
17 views

EM algorithm with constrained equation

I am reading a paper where author uses EM for the following equation to find the parameters $\theta$(and $\beta$) : $$ J=\sum_m \alpha_{m}\sum_i\sum_j w_{mij}\log\sum_k \theta_{ik}\beta_{mjk} $$ ...
0
votes
3answers
61 views

Hide and seek game

$A$ and $B$ go to the Senate to play a game of Hide-and-Seek. There are $100$ rooms in the Senate, and $B$ picks one of them and hides there till the game ends. $A$, at the beginning of every turn, ...
4
votes
1answer
40 views

Monopoly Game Statistics

I was playing a game of monopoly the other day, and in the course of strategizing I came up with the idea that how 'safe' you were in the game was a matter of what your expected income/outcome was as ...
0
votes
3answers
37 views

Is this a contradiction in probability?

In this question : $$ P_r(a\cap b)=P_r(a,b)=P_r(a)P_r(b)$$ However in this question: $$p(a,b) = p(a|b)p(b) = p(b|a)p(a)$$ Is this a contradiction as $P_r(a)P_r(b) \ne p(b|a)p(a)$ ?
0
votes
3answers
59 views

Stone, Paper, Scissors Game Winning Probability between two players in 1 match [closed]

I am required to find winning probability and algorithm of winning a game between two players in the above mentioned game. The catch is to find the winning stone, paper, scissor pattern so that ...
0
votes
1answer
17 views

A sample of 50 fluorescent light tubes from the SLT Company has a mean life of 20.5 hours and a standard deviation of 1.6 hours. Test: [closed]

A sample of 50 fluorescent light tubes from the SLT Company has a mean life of 20.5 hours and a standard deviation of 1.6 hours. Test: i. At the 1% level whether the sample comes from a population ...
0
votes
1answer
24 views

Binomial probability football betting game outcomes [closed]

There are normally three possible outcomes on a football match say home team win (1), draw (X) or visiting team or away win (2). If i have one game to bet, to cover all possible outcomes i will have ...
0
votes
0answers
66 views

Why doesn't combinatorics work here?

A while ago I asked one-to-one in combinatorics and then using one-to-one I'll repeat my answer here: There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are ...
-4
votes
1answer
66 views

what is the probabilty that sum of two random numbers between A and B is less than third number C [closed]

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$? I have tried answering this question using graph method to find the area under the ...
0
votes
2answers
27 views

Having a value, calculating the probability of the number of times a die has been rolled

If I enter a friend's house and he has rolled a dice (d6) on a table which has a value $1$, he asks me the following, "Do you think I rolled this dice once and got $1$ or do you think I rolled it ...
1
vote
0answers
31 views

How to prove: $E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$?

How to prove: $$E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$$, where $k$ is a positive integer, $x,y$ are fixed vectors, each entry in $w$ i.i.d. follows from an standard norm ...
0
votes
0answers
30 views

Perturbed density of eigen-states of a 3 diagonal matrix

How does the density of eigen-states ($D(\lambda)$ is defined as $D(\lambda) d\lambda$ = Number of states in the range $\lambda ... \lambda + d\lambda$) of the following tridiagonal matrix ($A$) ...
1
vote
1answer
30 views

Bivariate Probability PDF

If the joint probability density function of $X$ and $Y$ is given by $$f(x, y) = 24y(1 - x - y)$$ for $x > 0, y > 0, x + y < 1$, and $0$ elsewhere (A) Sketch Support $(X, Y)$ and find $P(X ...
6
votes
1answer
135 views

How to compute an expected value in shorter ways (when taking all possibilities into account isn't plausible.)

There is this question on which I have been spending a lot of time, trying to understand how to compute an expected value in a comprehensive way, as sorting out all the possibilities doesn't seem like ...
1
vote
1answer
37 views

Basic Probability Law Proof

I'm trying to show that $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Here's my attempt: Proof: There are two cases. Either $$A \cap B = \emptyset$$ or not. Suppose the former. Then $A$ and $B$ ...
-1
votes
1answer
40 views

Testing p(a|b) using 2 dice

p(a|b) = p(a,b) / p(b) Working this out Given two fairly sided dice each thrown once what is the probability of 1 or 2 in either dice ? ...
0
votes
1answer
33 views

Choosing suits of cards in a row

Three cards from a standard deck are dealt. What is the probability that the first is a heart, the second is a spade, and the third is another heart? I have figured out so far that you can use ...
0
votes
1answer
28 views

Fubini for Principal Value integrals in probability

Take a random variable $X$ with distribution function $F(x)=\mathbb{P}[X\leq x]$ and characteristic function $\phi_X(t)$. Then one can write ...
3
votes
2answers
139 views

Why are these following variance and expected value computations legitimate?

I spent over an hour of my exam's given time to calculate the variances and expected values as given here: Let $p,q\in (0,1)$. The number of costumers entering a supermarket is a r.v. $X$ with ...
2
votes
2answers
58 views

Strong and weak laws of large numbers

Let $X_1,X_2,\ldots$ be a sequence of random variables. Weak (strong) law of large numbers states that: If $X_1,X_2,\ldots$ are i.i.d. RVs and they have finite expectation $m$, then ...
2
votes
0answers
43 views

Probability ( Letters and Envelopes ).

A secretary types three letters and the three corresponding envelopes. In a hurry , he places at random one letter in each envelope. We need to find the probability that at least one letter is in the ...
0
votes
1answer
12 views

Random sampling-level of significance

Random samples of house selling prices are obtained from the north and south regions of a country. The results are summarized below: ...
2
votes
1answer
51 views

birthday problem - which solution for expected value of collisions is correct?

I am trying to understand the difference of the two solutions for the expected value of collisions for the birthday problem: http://math.stackexchange.com/a/35798/254705 derives the following ...
0
votes
1answer
21 views

Number of trials required to get value in a certain range

We do $M$ trials. In each trial the result is a uniform RV in $[0,1]$. What is the minimum no of tosses needed to be $90\%$ sure of getting a value in range $[0.8, 0.9]$. I figure the answer is $9$ ...
5
votes
3answers
99 views

How many ways to arrange the flags?

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements ...
0
votes
0answers
30 views

Is $E[E[Y_tZ_t|Y_t]|\mathcal{F}_{t-1}] = E[Y_tZ_t|\mathcal{F}_{t-1}]$ where $\mathcal{F}_t$ is the natural filtration process

As the questions is stated in the topic. Let $Y_t$ and $Z_t$ be discrete time-dependent random variables, and $\mathcal{F}_t$ is the natural filtration such that we know everything that has happened ...
1
vote
1answer
34 views

About 'Marcinkiewicz–Zygmund inequality'

Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki ...
1
vote
1answer
32 views

Comparison between two binomial lotteries

I've banging my head on the wall for the last two days with the problem below. Consider two lotteries based on two distinct binomial distributions, $B\left(n_{1},p\right)$ and ...
-1
votes
0answers
15 views

Taking a sample space for the total number of all possible outcomes when a pair of dice is flipped, [closed]

Taking a sample space for the total number of all possible outcomes when a pair of dice is flipped, find on a single flip of a pair of dice the possibility of obtaining A: a sum of ten B: an odd ...
0
votes
0answers
17 views

Normal approximation to Poisson-binomial and beta-binomial distributions

I am looking for normal approximations (preferably without any extra expansion terms) for Poisson-binomial and beta-binomial distributions. In particular, something better than the standard normal ...
0
votes
3answers
35 views

Bivariate Random Variable

Let $(X, Y)$ be a bivariate random variable with support $$S = \{ (x, y) \mid 0 < x < 7, x < y < x + 2 \}$$ and its joint pdf $f(x, y) = 1/14$ for $(x, y) \in S$. (A) Find the ...
0
votes
2answers
28 views

Probability ( Deck of Cards ).

Suppose 5 cards are taken at random without replacement from a normal pack of cards. We need to find out the probability of getting exactly 3 of a kind (the other 2 are distinct and are of different ...
3
votes
1answer
42 views

One-to-One correspondence in Counting

I have a confusion on the one-to-one correspondence in combinatorics. Take the problem: In how many ways may five people be seated in a row of twenty chairs given that no two people may sit next ...
0
votes
2answers
42 views

A problem on normal distribution.

The lifetime of a component in a computer is advertised to last for $500$ hours. It is known that the lifetime follows a normal distribution with mean $5100$ hours and standard deviation $200$ ...
1
vote
1answer
43 views

Dirichlet distribution when parameters $\rightarrow\infty$

I am reading a paper where they model a $\overrightarrow{\pi}$ random vector which is Dirichlet distributed in this way: $$ \overrightarrow{\pi}|\alpha\overrightarrow{w}\sim Dir(\alpha w_{1}+1, ...
1
vote
2answers
28 views

how manys ways are there if the order is taken into account?

Three candidates are selected from a certain number of interviewess. if the order is not taken into account, the number of ways the candidates can be chosen is 35. how manys ways are there if the ...