# Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

17,685 questions
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### Matching red to blue dots

I have two red points, $r_1$ and $r_2$, and two blue points, $b_1$ and $b_2$. They are all placed randomly and uniformly in $[0,1]^2$. Each dot points to the closest dot from another colour; closest ...
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### Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
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### Random sum in coupon collection

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
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### Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
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### Cramér's Model - “The Prime Numbers and Their Distribution” - Part 1

I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ...
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### Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that ...
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### What is the Probability of Transmission Between Two Nodes in a Neural Network?

I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is 1/N, meaning that if node n has fired ...
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### Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
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### At what rate does the entropy of shuffled cards converge?

Consider a somewhat primitive method of shuffling a stack of $n$ cards: In every step, take the top card and insert it at a uniformly randomly selected one of the $n$ possible positions above, between ...
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### Is this question solvable? $2$ non-linear equations and the proof that the solution is unique

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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### Generating function for number of $r$-disjoint subsets each of size $k$

Fix $n, k$. Then $$C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!}$$ is the number of ways to form $r$ disjoint subsets each of ...
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### Calculating growth rate of a population of Minecraft chickens

I have a rather strange question (for this Stack Exchange anyway). It felt too mathematical to ask elsewhere. If this is out of place here, please let me know. A chicken in Minecraft lays eggs; ...
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### How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
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### Asymptotic value of card drawing game

A deck consisting of $r_0$ red cards and $b_0$ black cards is randomly shuffled. The host turns up the cards one at a time; if it is red, you get $\$1$; otherwise you pay the host$\$1$ (and you're ...
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### Random graph connectivity, and the existence of isolated vertices

Here $G_{n,p}$ represents the Erdős-Rényi random graph model, where the graph has order $n$ and each edge is added independently with probability $p$. I am faced with proving the following claim: ...
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### Does this calculation have a name, or a generic formulation?

Background Informatiom I would appreciate help in identifying or explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: Sample from the distribution of each of $i$ parameters, ...
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### Probability that no team in a tournament wins all games or loses all games.

Five teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $1/2$ probability of winning any game it plays. Find the probability that no ...
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### Distribution of $i$th largest entry in multinomial sample

I have a $k$-class multinomial distribution with vector of probabilities $(p_1, p_2, \ldots, p_k)$, from which I draw a size-$N$ sample $(c_1, c_2, \ldots, c_k), \sum\limits_{s=1}^k c_s = N$. Assume ...
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### Probability that at least one of four hands missing at least one suit

Deal each of four players a 13-card hand at random. What is the probability that at least one of the four hands is missing at least one suit? Let $A_i$ mean that player $i$ is missing at least one ...
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### Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
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### How do Kolmogorov 0-1 law and CLT imply normalized sample mean doesn't converge in probability nor a.s.?

From WIkipedia the central limit theorem states that the sums Sn scaled by the factor $1/\sqrt{n}$ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-...
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### Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims that ...
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### Master Probability course question

A point of the lattice $\mathbb{Z}^3$ in $\mathbb{R}^3$ is painted white if at least one of its coordinates is odd. An ant is moving in $\mathbb{R}^3$. At each integer time $t$ the ant is at a point ...
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### Flies in a cube

Two flies sit in the corners $(1, 1, 1)$ and $(n, n, n)$ of a discrete cube $M^3$, where $M=\{1,\ldots, n\}$. After every unit of time both flies decide randomly and independently of each other ...
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### Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random Variable)...
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### Poisson: Conditional Probability on Pizza order

I am not sure about my answer. In particular, part b of the following question. Pizza orders arrive according to a Poisson process of rate 20 per hour. Orders are independently for a vegetarian ...