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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14 views

Probability of getting 2016.20162016…to infinity after dividing

Dividing 2240000/1111 seems to give 2016.20162016... to infinity. That is, 2016 keeps repeating. Can someone please tell me what the probability of this is? Thank you!
0
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0answers
5 views

Probabilistic least upper bound function

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
5
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1answer
41 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 ...
1
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2answers
39 views

Find the probabilty of 25 random people

X is the weight of one person, $X \sim N(\mu =78,\sigma =13.15 )$. If I choose randomly 25 people, what is the probability that the average of their weights will be $86$ ? I define $\displaystyle Z =...
0
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1answer
22 views

Conditional probability by joint probability

I have the joint pdf $$f(x,y)=\frac{1}{5}(3xy^2+2x^3y);0<x<1;0<y<2$$ and I have to calculate $$P(\frac{1}{2}<Y|X<\frac{1}{2})$$ I have found that $$f_{X}(x)=\int_{0}^{2}\frac{1}{5}(...
0
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1answer
25 views

Are $X_1$ and $X_2$ independent?

Let $X=(X_1,X_2)$ be an absolute continues random vector with the density function $f_X(x_1,x_2) = \left\{ \begin{array}{ll} \frac{2}{3}x_1+\frac{4}{3}x_1 x_2+\frac{2}{3}x_2, & \mbox{for } (...
5
votes
2answers
62 views

Probability of choosing $n$ numbers from $\{1, \dots, 2n\}$ so that $n$ is 3rd in size

We uniformly randomly choose $n$ numbers out of $2n$ numbers from the group $\{1, \dots, 2n\}$ so that order matters and repetitions are allowed. What is the probability that $n$ is the $3^{\text{rd}}$...
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0answers
36 views

How to choose between two options with a biased coin

We would like to choose between theatre and cinema by tossing a coin. Unfortunately the only available coin we have has probapility of heads $p\ \left(\dfrac{1}{2}<p<1\right)$. How could we use ...
2
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1answer
35 views

If we've got 10 coupons, what is expected number of different ones if there are 25 different types

I can't figure out this problem : There are 25 different types of coupon, all equally probable to get. If we have got 10 coupons, what is expected number of different coupons between them? ...
1
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1answer
37 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
0
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3answers
48 views

True or false:if $A\subset B$, then $P(A)<P(B)$?

They ask me if this statement is true or false, and explain why. They suggest I write an example showing why it is false or true. The statement is: if $A\subset B$, then $P(A)<P(B)$. What I ...
0
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0answers
16 views

Bounding Probability Distribution

I have the following problem. Let $X$ be a continuous random variable with image $[0, b]$ for some finite $b>0$. So we have finite moments, $\mathbb{E}[X^n]$. I am hoping to say something about the ...
2
votes
1answer
37 views

Which has higher variance, coin toss vs dice roll?

Dusting off some high school stats and getting confused over the following: Two betting games: Pick right side of coin, even-money bet ($p = 0.5$, $q= 0.5$), Pick right value in a 10-sided ...
3
votes
1answer
34 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 ...
0
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0answers
11 views

Is there any example of a Markov chain (discrete) with limit distribution (discrete) of heavy tail?

Is there any example of a Markov chain with limit distribution (discrete) of heavy tail? In other words, a Markov chain whose limit distribution has infinite second moment?Already, thanks for the help!...
5
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1answer
1k views

Fingerprint match probability

I am trying to use the formula for the birthday paradox as a reference to figure out an equation that represents the probability of a fingerprint match. Here's the equation for probability of a ...
1
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1answer
667 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
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0answers
21 views

Looking for interesting applications of Ergodic Theory in mathematics [on hold]

I would like a list of nice applications of ergodic theory, in mathematics or probability theory. I think there are several applications of ergodic theory to continued fractions, number theory in ...
1
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0answers
31 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
2
votes
2answers
74 views

I choose three random integer point in origin $|x|, |y|\leq r$. plane, what probability to this point creates a right triangle?

I wont to choose three random integer point in origin $|x|\leq r, |y|\leq r$ at plane as $(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})$. What the probability that this three point create a right triangle ...
0
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0answers
24 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
2
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1answer
339 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
0
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0answers
15 views

Data transmission process PDF

Given the quasi-defined data transmission random process: $X(t) =\sum_{n=-\infty}^{+\infty} a_n \pi_T(t - nT)$ where $a_n$ are statistically independent RVs that can either assume the value 0 or 1 ...
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0answers
12 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
1
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1answer
13 views

The intuition behind conditional probability and independence in the case of different sample space

I came up with this question when doing this problem: In throwing a pair of dice, let A be the event that "the first die turns up odd", B the event that "the second die turns up odd", and C the ...
5
votes
2answers
88 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 ...
0
votes
1answer
28 views

Probability of a permutation for inversions

Sample space for following problem is S4. And the probability $p(\sigma)$ of a permutation is $\alpha$ times the number of inversions of $\sigma$ for suitable $\alpha$. We have to find the value of $\...
2
votes
2answers
26 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 ...
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2answers
61 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? I have looked at similar past questions such as Expected Number of Coin Tosses to Get Five Consecutive Heads ...
0
votes
1answer
31 views

The minimum amount of tries for a matching game?

The game, match it up, where you've got 20 cards and each card has another to match up to which is the same, i was wondering, is there a least amount of tries you can complete this game in or is it ...
0
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1answer
47 views

Probability of skewed coin vs fair coin given conditions.

I'm solving the following problem: We have 2 coins, Coin A is heads with probability $\frac{1}{2}$ and Coin B is heads with probability $\frac{2}{3}$. We choose a random coin with equal chance of ...
0
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1answer
29 views

Unique draws from a pool, with replacement

I am attempting to find the probability of selecting $m$ unique items from a total of $n$ items with $k$ draws, with replacement. Let's denote this $P(n, m, k)$. Here's my thought process: $P(n, 1, k)...
0
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1answer
64 views

$A$ is a set containing $n$ distinct elements.A non zero subset $P$ of $A$ is chosen.

$A$ is a set containing $n$ distinct elements.A non zero subset $P$ of $A$ is chosen.The set A is reconstructed by replacing the elements of $P$.A non zero subset $Q$ of $A$ is again chosen.If the ...
0
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1answer
30 views

Conditional Probability given unlimited information?

We know that $P(A|B)$ is more accurate than $P(A)$ because we taken into account the information $B$. My question is if we know all the information contributing to "$A$ occurs" then should P(A|those ...
0
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1answer
26 views

Probability in the game Resistance

I was playing the game Resistance with a group of 10 people. In the game, people are given one of two "assignments". 6 people are given cards that tell them they are part of the Resistance. 4 people ...
0
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1answer
25 views

Finding the probability when knowing the number of successes after n attempts

Assuming I have made an experiment, and after n attempts I had k successes, how can I know what is my probability of succeeding in a single attempt? Just for some realism, the numbers I'm happening ...
0
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1answer
18 views

capture-recapture - Chance of tagging twice

A friend of mine asked me if I could solve this following problem, and it has turned out to be rather difficult (or I just cant find the answer) Let N be the total population. We take a samle of K ...
0
votes
2answers
28 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 ...
1
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2answers
35 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}=\...
0
votes
1answer
22 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
63 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
vote
1answer
34 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
votes
1answer
61 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
38 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
21 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 ...
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1answer
52 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
28 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 $f(x)$ ...
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1answer
53 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 ...
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2answers
27 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 ...