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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2answers
103 views

Guessing sum of random sub set of length x from one set

Given a known Set x . Given a know size y. User choose a random subset from x of size y. We need to guess the sum of the new subset. both x and y are less then 5000. After searching on net this ...
2
votes
2answers
864 views

name for a rational number between zero and one?

I'm searching for a unified name to convey for the concept that a number will always be between zero and one. Some info for context: in probability we've got a number between 0 and 1. Percentages ...
5
votes
2answers
690 views

What's the expected value of average absolute deviation from the mean of k randomly picked numbers?

Say we have to randomly pick k integral numbers out of n. The numbers are from the range < a; b >. What is the expected value of average absolute deviation from the mean for that random subset of ...
3
votes
2answers
189 views

Question about computing a Fourier transform of an integral transform related to fractional Brownian motion

I am trying to show an integral transform has a fixed point. Let $H \in (0,1)$ and consider the following integral transform whose kernel is the density of fractional Brownian motion: $$T_H f(x) = ...
3
votes
2answers
140 views

Reaction-diffusion equations and stochastic processes

The solution to the Fokker-Planck equation can be thought of as a macroscopic description of the dynamics of a diffusion process. Various results make this heuristic more precise - Ito integration, ...
24
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6answers
1k views

What are Belgium's chances on getting a medal at the 400m finals, given that they have 2 of the 8 athletes?

This is a question my wife and I pondered over at the breakfast table this morning: Belgium has two athletes in the finals of the 400m at the World Championship athletics this year. What are ...
2
votes
3answers
145 views

Why $E[X|X>1] = E[X] + 1$ for geometric random variable?

For a geometric random variable X, $P_X(k) = (1-p)^{k-1}p$, then $$E[X] = \sum_{k=1}^{\infty}k(1-p)^{k-1}p\;.$$ That is ($q = 1 -p$) $$E[X] = p + 2qp + 3q^2p + 4q^3p + \dotso $$ But ...
2
votes
3answers
606 views

Probability that two poker hands have no aces

The probability of no two hands from the same poker deck having no aces is $$\frac{{48 \choose 5} + {48 \choose 5} - {48 \choose 10}}{{52 \choose 5}}$$ I am not sure why this is the answer, as the ...
5
votes
2answers
159 views

Probability for rearranging balls in particular order

Given $N$ boxes, each containing one ball each say numbered as $B_1, B_2, B_3, \ldots, B_N$. We take all of these balls out and put them back in different boxes but not their original one. So, $B_1$ ...
2
votes
1answer
98 views

Base-rate fallacy?

I tried to make a counterexample that splitting data into sets can lead to the wrong conclusion and it seems to me to generate a fallacy just by splitting data into subsets. What I did was making up ...
2
votes
1answer
135 views

conditional expectation — a specific problem

Suppose that $X$ and $Y$ are independent random variables, $X$ is uniformly distributed on $[0,1]$, and $Y$ is uniformly distributed on $\{ 1,2,3\}$. How to get $E((X+Y)^2 | Y)$? It seems to me that ...
3
votes
3answers
975 views

How to obtain the Standard Deviation of a ratio of independent binomial random variables?

X and Y are 2 independent binomial random variables with parameters (n,p) and (m,q) respectively. (trials, probability parameter)
39
votes
4answers
2k views

What's the probability that a sequence of coin flips never has twice as many heads as tails?

I gave my friend this problem as a brainteaser; while her attempted solution didn't work, it raised an interesting question. I flip a fair coin repeatedly and record the results. I stop as soon as ...
1
vote
1answer
206 views

Depth distribution of normalized decision trees?

Lets work with the following inductive definition of a decision tree: 1) $\bot$, $\top$ are decision trees. 2) If $x_i$ is a variable and $T_0$, $T_1$ are decision trees then $(\lnot x_i \land T_0) ...
1
vote
1answer
350 views

Probability having to do with “Russian roulette”

A shotgun has 6 slots and there is 1 bullet in it. The first person pulls the trigger and nothing was fired. The second person pulls the trigger without shuffling the chamber. What's probability that ...
4
votes
3answers
730 views

Bounds on off-diagonal entries of a correlation matrix

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the ...
30
votes
4answers
3k views

Probability for the length of the longest run in $n$ Bernoulli trials

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, ...
2
votes
2answers
281 views

Limiting distribution of the sum of squared independent normals

Let $t\in[0,1]$ and suppose $Z_t$ is a standard normal random variable, such that $Z_t$ and $Z_s$ are independent if $s\neq t$. If we pick $0\leq t_1\leq \ldots\leq t_k\leq 1$ then $$X = ...
9
votes
2answers
341 views

Proportion of spanning trees in a network in a social media messaging context

Consider a graph, such as the following: I'm considering a model of message propagation (e.g. re-tweeting) in this network, starting from a root node (e.g. the node 1 in the lower-left). ...
4
votes
3answers
602 views

Throw a die $N$ times, observe results are a monotonic sequence. What is probability that all 6 numbers occur in the sequence?

I throw a die $N$ times and the results are observed to be a monotonic sequence. What is probability that all 6 numbers occur in the sequence? I'm having trouble with this. There are two cases: when ...
2
votes
2answers
262 views

probability terminology for parameter in a Markov process

Suppose $$P(\text{feature present at time} \ t \ \text{and} \ t+\Delta t) = \beta^{2}+\beta(1-\beta) \exp(\Delta t/\tau)$$ where $\tau = 1/(\pi_{01}+\pi_{10})$. What is $\tau$?
1
vote
2answers
1k views

Infinite coin tosses as tail event?

I was reading about Kolmogorov's zero-one law specifying: a certain type of event, called a tail event, will either almost surely happen or almost surely not happen I came to this example: ...
2
votes
1answer
193 views

An example of Markov-Feller chain with some properties

Let $X$ be a Polish space and $C(X)$ denote the space of all bounded and continuous functions on $X$. We consider a Markov chain $(\xi_n)_{n\geq 0}$ with transition probability $P:X\times ...
5
votes
1answer
306 views

Average area of choosing three points on a surface?

Assume I choose three random points on the surface of a sphere. What is the average area? (Each point is independently chosen relative to a uniform distribution on the sphere) Also, what would be the ...
1
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1answer
1k views

understanding Log likelihood

I have always had a problem of understanding the big picture of probability and get lost every time I am studying it. I am working on implementing a X-means algorithm for clustering data and the log ...
4
votes
1answer
231 views

An alternative solution to a probability problem

The text states: There are 3 seniors and 15 juniors in Mrs. Gillis’s math class. Three students are chosen at random from the class. What is the probability that the group consists of a senior ...
1
vote
2answers
2k views

Probability and Permutation with Repetition

The problem states: The letters of the word TOMATO are arranged at random. What is the probability that the arrangement begins and ends with T? I calculate n(S) to be 6! = 720 I calculate n(E) ...
0
votes
1answer
386 views

Probability accuracy measurement

Let's say that for each trial, there are two possible outcomes, A and B. Over time, A shows that it occurs 54% of the time, B is 46%. But there is another variable that affects the outcome of each ...
3
votes
0answers
90 views

Different definitions of e-property for Markov-Feller chains

Let $X$ be a Polish space. We consider a stochastic kernel $P:X \times \mathcal{B}_X \to [0,1]$ and the Markov semigroup $(P^{\;n})_{n\geq1}$ of iterations of $P$, which satisfy the Chapman–Kolmogorov ...
2
votes
2answers
128 views

“$1-P$” Probability Problem

A problem states: There are 8 boys and 12 girls in a chess club. Two boys and two girls are to be selected at random to represent the club at a tournament. What is the probability that Jacob ...
0
votes
1answer
60 views

The rough average of searches needed, to find key out of n keys

I've a question that has been bothering me regarding combinatorial searches, such as brute force of a key for an encryption. Lets say I have 2^32 keys in a key space that would need to all be ...
2
votes
1answer
96 views

Reasoning error in “maximin” type problem

I have a problem that I have approached two different ways. The approaches give me two different answers. I have matched the answers against a simulation, so I think I know which one is right. But, ...
0
votes
1answer
272 views

Probability density in kNN algorithm

I have been reading an implementation of the KNN algorithm to determine what is the probability that the price of an item A with certain attributes is between X and Y dollars. In order to find such a ...
3
votes
1answer
97 views

Rate of change of $\mathbb{E}X_{\frac{2^n}{2},2^n}$ as $n$ increases?

I am trying to get an equation that will show the rate of change of the expected value of $\frac{2^n}{2}$th lowest of $2^n$ draws from $X$ as $n$ increases (where $n >1$). Let's call the order ...
1
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0answers
65 views

Sure loss and comparison of marginal integrals. Looking for a proof or a counter-example

Consider the following integrals $$\int g(x,y) N(y;x,1) dy$$ and $$\int g(x,y) N(x;y,1) dx$$ where $N(x;y,1)$ is the Normal PDF of the variable $x$ with mean $y$ and variance $1$ ($x$,$y$ both real ...
1
vote
0answers
143 views

Birthday paradox with M shared birthdays [duplicate]

Possible Duplicate: Probability of 3 people in a room of 30 having the same birthday The birthday paradox is a common problem about the probability that any 2 people from a random set of ...
3
votes
1answer
511 views

Minimize Expected Value

Can anyone prove, or disprove with a counter-example, the following statement: The value of $\sum_{k=1}^n kp_k$ with $\sum_{k=1}^np_k = 1,\,$ is minimized if $p_1 \ge p_2 \ge \cdots \ge p_n$, and ...
3
votes
2answers
643 views

Brushing up on Linear Algebra and Probability

I am a programmer/analyst with limited (and pretty rusty) knowledge of math. "Just for the heck of it" I have decided to try my hand at Stanford's introductory course on Artificial Intelligence and ...
3
votes
1answer
211 views

Computing quadratic variation for stable Levy flights with $0<\alpha<2$?

The wiki page on semi-martingales states that Every Lévy process is a semimartingale. and that The quadratic variation exists for every semimartingale. Let $X_t$ be a stable Levy process ...
1
vote
1answer
78 views

A problem on probability

What is the probability that a license plate consisting of 6 distinct letters (A-Z) will have its letters in alphabetical order.
2
votes
0answers
138 views

How to model server load?

I want to model the server under load. I'm using following assumptions: The server serves only one request at a time, and all requests take him exactly 100ms to process. All requests that came ...
3
votes
1answer
149 views

Bayesian analysis of the Venice Doge elections

Does anyone know of a Bayesian (or a classical) analysis of the Venetian Doge election system? I am looking mainly for chances of subversion, chances for a candidate to be elected at each stage, or ...
3
votes
1answer
127 views

The limit of the expectation of the top half+1 order stats of $n$ draws of $X$ as $n\to\infty$?

Can anyone help me compute the limit of the average of the top half +1 of order marginal order distribution of $n$ draws from $X$, as $i\to\infty$? Specifically, the limit as $i\to\infty$ of ...
1
vote
1answer
102 views

Given a randomness source with min-entropy $k$, what is the expected number of samples without repetitions?

Given a randomness source $D$ with min-entropy $k$, we should expect that we will have to take more than $2^{k/2}$ samples before we see a repetition (claim from ...
3
votes
2answers
164 views

Markov Chains and Linear Transformations

I just have a quick question about Markov Chain and linear algebra. Background. Let $\{M_n: n= 0, 1, 2, \dots \}$ be a Markov Chain. We can represent the transition probabilities $_{n}Q^{(i,j)}$ in a ...
1
vote
4answers
289 views

Can a vertical line be a valid PDF?

Consider a random variable X that always takes a single value, c. I would think that a valid PDF for X would be $$f(x) = \begin{cases} 1/c & x = c \\ 0 & \text{Otherwise}\end{cases}$$ ...
3
votes
2answers
151 views

Limit of a nested probability function?

Let $F_0(x) =x$. Then, let $F_i(x) = (1-(1-F_{i-1})^2)^2$ where $i>0$. Is there a way I can calculate $F_{\infty}(x)$? (Terminology-wise, would I say that is a limit of $F_i(x)$ as $i \to \infty$ ...
1
vote
1answer
97 views

Order statistic distances as a function of moments?

Let $X$ be a random variable defined over $0-1$ and let $X_1 . . . X_4$ be the order statistics of $4$ draws from $X$ (where $X_1$ is the minimum). I am looking to see if I can identify any general ...
5
votes
3answers
5k views

How many rolls do I need to determine if my dice are fair?

Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die? Note that I will be actually manually rolling physical ...
11
votes
3answers
593 views

A question about Poker (and probability in general)

Okay, so I've been thinking about this question for a long time, and I'm starting to think that there isn't an answer. So please read the question, and if there is an answer, tell how you came to it, ...