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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0
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1answer
19 views

Relationship Between $\mathbb{E}$(time) and $\mathbb{E}$(Repetition)

Consider aa Stochastic Process with Expected value of time of occurring =T (less than infinity). Can we deduce that Expected value of number of occurrences until time T is equal to 1?? If not, in ...
3
votes
1answer
54 views

The probability two balls have the same number

Suppose I have $10^6$ jars, and $k$ balls are randomly and independently placed in each jar. I am given that the probability that there exists a jar with 2 balls is approximately $50\%$. Then $k$ is: ...
1
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1answer
33 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\displaystyle\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\displaystyle\frac{...
2
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1answer
57 views

Using the Central Limit Theory to solve $\lim_{n\rightarrow \infty} \mathbb{P}(n-\sqrt n \lt X_1+X_2+\cdots+X_n\lt n+\sqrt n)$

$X_1,\ldots,X_n$ are independent random variables that are uniformly distributed between 0 and 2. What is: $$\lim_{n\rightarrow \infty} \mathbb{P}(n-\sqrt n \lt X_1+X_2+\cdots+X_n\lt n+\sqrt n)$$ ...
2
votes
3answers
36 views

Expected number of married couples chosen out of 50 different people

I've encountered this problem, and would like to know if my approach is right. We select 10 people out of a group of 25 married couples, what is the expected number of married couples chosen? ...
0
votes
2answers
20 views

Conditional Probability: Birth rank of children in randomly chosen families

(BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
1
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1answer
49 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
1
vote
1answer
27 views

Is this definition of a continuous random variable correct?

I was a bit puzzled, because it seems like a discrete random variable would also satisfy the following definition: Definition: A random variable $X$ is continuous if there is a function $f(x)$ ...
3
votes
1answer
29 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
1
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1answer
46 views

Real life illustration of the fact that rationals have measure zero

I wonder if there's any real world phenomenon that reflects the mathematical fact that $\Bbb Q^k$ has Lebesgue measure zero in $\Bbb R^k$, or put another way, the likelihood that we get a rational ...
0
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2answers
26 views

Expected value and variance of a random variable, defined as the largest of $6$ randomly drawn numbers

Let each of the numbers from $1$ up to $49$ be written on a ball, and let all these balls be contained in a box. From this box, we randomly draw exactly $6$ numbers (without putting them back, so we ...
1
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1answer
6 views

Independent events from any other

In $(\Omega,\mathcal{F},P$) probability space, how can I show that $\forall A\in \mathcal{N}=\left\{ A\in\mathcal{F}: \mathbb{P}(A)=0,or\,\, \mathbb{P}(A)=1 \right\}\Rightarrow$ $\forall E\in\Omega$...
2
votes
1answer
20 views

References for the applications of probability in gambling

The intuition behind many theorems in probability comes from gamblers' games. I would like to know if there are any books or articles which cover some such connections between probability and its ...
2
votes
1answer
34 views

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$?

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$? For context, I'm re-reading Kallenberg and in Chapter 3, on page 49, in his proof of Lemma ...
0
votes
1answer
22 views

Sum of Random Variables i.i.d. with $\mathbb{E}[|X_n|]=+\infty$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $\mathbb{E}[|X_n|]=+\infty,\forall n$. Prove that $\sum_n \mathbb{P}[|X_n|>kn]=\infty$ with $k\...
0
votes
0answers
25 views

Finding expected value of $\mathbb{E}[X^{2}Y]$ [on hold]

$X,Y$ are random Variables. $X = 2,4 ; Y = 1,3,5;$ I have to find $\mathbb{E}[X^{2}Y]$. I know that - $\mathbb{E}[g(X)] = \sum_{r} g(r)\mathbb{P}(x=r)$ But I don't know what the formula for $\...
1
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1answer
22 views

Finding the expected value of the third suitable cedar cone

The cones of the Lebanese cedars have (widest) circumferences that are Gamma-distributed random variables with mean of $5.40\ cm$ and variance of $3.24\ cm^2$. A botanist is interested in ...
2
votes
1answer
28 views

Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
-1
votes
1answer
24 views

Problem involving sequence of random variables on probability space [on hold]

How do I construct (and prove that) an example of a sequence of random variables $\{X_n\}_{n\, \ge\, 1}$, on an appropriate probability space, for which $X_n$ converges to $0$ in $L^r$ for all $r > ...
0
votes
1answer
19 views

What is the difference between a reversible markov chain and a reversible in equilibrium markov chain?

In the text I'm using it says: Let X = {$X_n : 0 \leq n \leq N$} be an irreducible Markov chain such that $X_n$ has the stationary distribution $\pi$ for all $n$. The chain is called reversible if ...
1
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0answers
29 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
1
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1answer
31 views

If the diameters of ball bearings are normally distributed, determine the percentage with diameters between $0.610$ and $0.618$ inches.

If the diameters of ball bearings are normally distributed with mean $0.6140$ inches and standard deviation $0.0025$ inches, determine the percentage of ball bearings with diameters Between $0.610$ ...
0
votes
1answer
26 views

Probability: Application Of “Expected Value”

$\newcommand{\P}{\mathbb{P}}$$\newcommand{\E}{\mathbb{E}}$So, I was learning expected value today and I'm trying to understand the significance of calculating this term "Expected value". In this ...
1
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1answer
26 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
3
votes
1answer
24 views

Figuring out probability of dice with least amount of questions

Given $n$ dice, each with $k$ faces numbered $1,\dots,k$, you're allowed to ask me what the probability of some event happening is (a subset of all the possibilities and I give a number). What ...
0
votes
1answer
33 views

5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : ...
4
votes
2answers
66 views

Is there a quick way to justify that this elementary probability is equal to 2/3?

I just solved this problem with the conditional probability formula and after a while the answer was surprisingly 2/3. I believe there must be a tricky short way to calculate it. Can somebody help ...
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0answers
29 views

$\newcommand{\P}{\mathbb{P}}$ If a,b are independent, can I prove that $\P(ab|c)=\P(a|c)\P(b|c)$ and $\P(a|bc)=\P(a|c)$? [on hold]

$\newcommand{\P}{\mathbb{P}}$ If a,b are independent, can I prove something below: A. $\P(ab|c)=\P(a|c)\P(b|c) $ B. $\P(a|bc)=\P(a|c)$ thanks
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0answers
23 views

conditional expected value and not mutual indipendent events [on hold]

$\newcommand{\P}{\mathbb{P}}$ Let be $E,G,H$ pairwise independent events but not mutual (e.g. $P(E\cap H)=\P(E)\mathbb{P}(H),\,\P(G\cap H)=\P(G)\P(H), ...but \,\P(E\cap G\cap H)\ne\P(E\cap G)\P(H)$ ...
5
votes
1answer
28 views

Difficulty understanding step in Kac's proof of Feynman-Kac Theorem

I am trying to understand a proof of the Feynman-Kac Theorem, as set out in Mark Kac's 1949 paper 'On Distributions of Certain Wiener Functionals'. Kac defines a series of independent and ...
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0answers
19 views

Game theory: how is law of large number applied here?

This is a claim rephrased and lifted from from Herbert Gintis' book "Game Theory Evolving" Pg187 Consider an evolutionary game with $n$ pure strategies $i = \{1, \ldots, n\}$, and time periods $t ...
0
votes
2answers
97 views

Probability Of four events

I have an issue with calculating probability of union of four events, formula listed below \begin{align*} P(A \cup B \cup C \cup D) & = P(A) + P(B) + P(C) + P(D) - P(A \cap B) - P(A \cap C)\\ &...
0
votes
1answer
22 views

Is there a name for the distribution of this CDF function?

CDF: $F(x) = (1-e^{-a \cdot x^2})^{\frac{b}{c-x}}$ where $a,b,c$ are positive constants, and $x \geq 0$. Can any body give some advice on how to analyze the mean, variance or any other properties ...
0
votes
1answer
486 views

probability - ice cream flavours

Of the $50$ ice cream flavours at J.P. Lick’s, $10$ of the ice cream flavours have a vanilla base (as opposed to chocolate or some sort of other flavour base). Of the $50$ ice cream flavours, $15$ ...
-1
votes
1answer
30 views

does this converge? [on hold]

If I have $$X_n=\begin{cases}x_n & p_n\\ 0 & 1-p_n \end{cases}$$ and I know that $x_n$ converges to $0$ as $n$ tends to $0$, can I say that $X_n$ converges to $0$ almost sure?
1
vote
1answer
23 views

Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
0
votes
1answer
39 views

What is the average and variation of $20$ dices?

If I roll a dice the average is $E(X) = (1+2+3+4+5+6)/6 = 7/2$ and $$E(X^2) = (1+4+9+16+25+36)/6 = 91/6$$ $$VAR(x) = E(X^2) - (E(X))^2 = 91/6 - 49/4 = 35/12$$ Now the question is: How I can find ...
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votes
2answers
49 views

Expected number of tosses to get 3 consecutive Heads [on hold]

I have a fair coin. What is the expected number of tosses to get three Heads in a row? Apparently, I could not find any completely clear approach/solution to this problem.
1
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2answers
25 views

Permutations in an Infinite List of Random Numbers

In an infinite list of random numbers from a to b, prove that in this list, there are all possible permutations of n numbers from the list, where n can be any number. Here are some versions of the ...
1
vote
1answer
17 views

Finding the method of moments estimator for the Uniform Distribution

Let $X_1, \ldots, X_n \sim \text{Uniform}(a,b)$ where $a$ and $b$ are unknown paramaters and $a < b$. (a) Find the method of moments estimators for $a$ and $b$. (b) Find the MLE $\hat{a}$ ...
6
votes
1answer
187 views
+200

Is there any probability model for multi-stage motion of an object such as this.

I have this following case (please refer to attached pic below) where a particle is resting on the ground and it needs a minimum amount of force (Fmin) to reach from one level to the next level. But ...
1
vote
1answer
37 views

Simple problem in probability

You have 100 lightbulbs. Every lightbulb is either functioning or not. You test 20 of them, and all of the 20 are functioning. What is the probability that 10 of the 100 lightbulbs do not function? ...
1
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0answers
24 views

how many samples needed to obtain an estimate with a given confidence interval

Suppose an urn contains N balls of different colors. I do not know the colors nor the distributions, and I wish to determine the fraction of red balls in the urn, (R/N), to within p% with C confidence....
1
vote
1answer
23 views

Betting ended after nth round.Find the sum of money NOT WON?

Rahul and Vijay are playing a game with 12-sided die,where both of them lay bets on outcomes of roll of die.They start betting Rs 5 each on first round of the game and the amount bet in each ...
1
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0answers
101 views

Switching independent experiments: Does $I_A(X_1,\dots,X_n,Y_{n+1},Y_{n+2},\dots)$ converge almost surely?

Suppose there is a sequence of independent random variables $X=(X_1,X_2,\dots)$ with $X_i$ taking values in some arbitrary measure space $E_i$. Now there is a second sequence $Y=(Y_1,Y_2,\dots)$ ...
1
vote
2answers
475 views

A conditional probability problem on coupon collection

Suppose that there are $n$ types of coupons, and that the type of each new coupon obtained is independent of past selections and is equally likely to be any of $n$ types. Suppose one continues ...
3
votes
1answer
91 views

Probability of getting loops

You are given $3$ bits of lace, if ends are tied together at random, what is the probability that you end up with $2$ loops? Generalise this for $n$ bits of lace. Ok so clearly I have 6 ends to play ...
1
vote
2answers
49 views

There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. What's E[X=the number of empty boxes?]

There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. The red balls are placed in boxes 1-10, blue balls are placed in 1-6. What is the expected ...
1
vote
0answers
18 views

Calculate probability of an event occurring exactly once in an arbitrarily selected year, given mean annual occurrences over a century

If an event occurs an average of 0.6 times/year over a century, what is the probability that it occurs exactly once in a randomly selected year? I was able to find p(occurs exactly once in a given ...