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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1answer
18 views

Probability of sums with 6 dice

You roll six independent fair dice. What is the probability that their sum is divisible by 6? I don't really know where to start. Does the ordering of the dice matter? (1,2,2,2,2,3) vs (3,2,2,2,2,1)....
1
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2answers
15 views

Probability of increasing order permutation

Suppose I have n elements. What's the probability of a permutation such that the first half is increasing and second half can be ordered without any constraints? (A permutation can only have distinct ...
0
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0answers
15 views

The probability for a team to win the cup in a championship. A simplified case.

Hypotesys: For each two national football teams $i$ and $j$ that play against each other the probability of victory for $i$ is defined as: ${{p}_{i}}=\frac{{{N}_{i}}}{{{N}_{i}}+{{N}_{j}}}$, where $N$...
-2
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0answers
7 views

Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
1
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1answer
57 views

Someone Ripped Me Off, Please Help Calculating Odds!!

I'm protesting a state contract, and one of the grounds for protest is that someone stole material from a past proposal my company submitted, and is representing it as their own. Besides leaving our ...
0
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1answer
18 views

Independence of linear combinations of random variables

This seems like a straightforward question, but I'm having trouble finding anything on it. Suppose we have a set of random variables, $X = (X_1,...,X_p)$ (the components of which may not be ...
2
votes
2answers
43 views

When should we consider objects as distinguishable in probability?

Example : Why is the probability of getting a sum of 12 when we roll two fair dices is 1/6 whereas probability of getting 5 is 2/6 . When we labeling the dice red and green in our head , isn't (6 ...
0
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0answers
15 views

Property of Conditional Probability Multiple Additive Variables

I am working to explore a property of conditional probability that I could use some assistance with. The general problem can be stated as $$\Pr(a > b \mid z^*)$$ where $z = z^* + u$. So then $$\...
0
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1answer
29 views

Probability that two random sets have at least one element in common

We consider a finite field $\mathbb{F}_p$, where $p$ is a large prime number, i.e. $p$ is a 128-bit number. Person $A$ chooses $d$ values drawn uniformly at random from the field. Also, person $B$ ...
2
votes
4answers
36 views

Probability of a dice launched three times

A dice is launched three times. What is the probability to obtain three even numbers ? I've solved this problem calculating the number of total results: $$u=D'_{6,3}=6^3$$ and the number of ...
0
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0answers
18 views

Conditional expected value not mutually indipendent sets

Let be $E,G,H$ pairwise independent events but not mutual (e.g. $\mathbb{P}(E\cap H)=\mathbb{P}(E)\mathbb{P}(H),\,\mathbb{P}(G\cap H)=\mathbb{P}(G)\mathbb{P}(H), ...but \,\mathbb{P}(E\cap G\cap H)\ne\...
3
votes
1answer
29 views

Maximize sum with no two consecutive variables

Random variables $x_1,x_2,\dots,x_{100}$ are drawn independently from the uniform distribution over $(0,1)$. After knowing the values, we are allowed to choose a subset of them as long as no two ...
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2answers
42 views

What is the probability of random walking ant to be at a position after some finite steps on an infinite grid? [on hold]

Is it even calculable? What if the grid is infinitely dimensional? Lets say that it is a simple random walk, and probability to move to any neighboring position is equal, but other types are also ...
1
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2answers
32 views

Why does a null-hypothesis have to have a definite value?

In hypothesis testing, why does the null hypothesis (H_0) have to have one defined value?
0
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2answers
53 views

Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
1
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0answers
20 views

Differences of Markov chain is Markov

In my studies of Markov chains, I was tackled with this tough problem: Let $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain with transition probabilities satisfying $ | i-j | > 1 \to ...
0
votes
0answers
12 views

Forecasting for a non-steady state problem

Let say, at each discrete time step, $t_i$, we can forecast of a specific event's occurring rate, $I_{trans}$ by following formulation: $$I_{trans}(t) = I_{ss}. \sqrt{(\frac{A\tau}{t})} .e^{-\frac{B\...
8
votes
1answer
53 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
-6
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1answer
33 views

Finding the number of possible shortest ways. [on hold]

Find the number of possible shortest ways from A to B.
0
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3answers
34 views

Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
2
votes
1answer
32 views

Arrange 18 pips on a die with at least one 0 side to maximize the probability that 5 rolls sum to 13 or more.

You are arranging pips on a standard 6-sided dice. Rules: At least one side must be left blank at 0. The average roll must be 3 (so, you have 18 pips to distribute among five sides). You want to ...
0
votes
2answers
23 views

Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
3
votes
1answer
29 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
0
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0answers
6 views

Drift analysis of an absorbing Markov chain

Consider a set $S$, and suppose we have a sequence of random subsets $$ \zeta_t = \{x_1, \dots, x_n\} $$ for $x_1, \dots, x_n \in S$. We do not know with which probability density the points of each $\...
0
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0answers
36 views

Probabilistic Modeling Parameters Request

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,...
1
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2answers
48 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
votes
1answer
27 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
votes
1answer
27 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 } (...
0
votes
3answers
54 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 ...
2
votes
2answers
56 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? ...
0
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0answers
20 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 ...
6
votes
3answers
83 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 $3^{\text{rd}}$...
0
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0answers
13 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!...
2
votes
1answer
46 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 ...
1
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0answers
36 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 $...
1
vote
1answer
52 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 ...
0
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0answers
31 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, ...
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0answers
14 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 ...
2
votes
1answer
16 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 ...
0
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0answers
17 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 ...
0
votes
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 ...
5
votes
1answer
45 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 ...
2
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2answers
27 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 ...
0
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2answers
31 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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0answers
25 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 ...
-1
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1answer
33 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
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2answers
28 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
votes
2answers
62 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 ...