0
votes
1answer
31 views

Probability for incomplete information

Let's say there are 10 teams: A-J. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be calculated. Not all teams participate in each game. ...
0
votes
1answer
33 views

How to calculate probability that a team will win

Let's say there are 10 teams: A-J. Each team always participate in each of the game. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be ...
3
votes
3answers
267 views

2 of 3 dice are selected randomly and thrown. What is the probability that one of the dice shows 6

1 red die with faces labelled 1, 2, 3, 4, 5, 6. 2 green dice labelled 0, 0, 1, 1, 2, 2. Answer: 1/9 Please can you show me how to get the answer. I'm confused about joining the events of choosing 2 ...
5
votes
4answers
133 views

Probability that a word contains at least 3 same consecutive letters?

Assume we have a word of length $n$ and an alphabet of length $26$ (the small letters a through z, if you want so. How likely is it that this word contains at least $k := 3$ consecutive letters of ...
0
votes
0answers
12 views

exponential bound on the number of possible clusters at $0$ in $\mathbb{Z}^d$

Let us say that $\mathbb{Z}^d$ is given the usual lattice structure as a graph. I want to know the number of connected induced subgraphs of size $k>0$ that include the vertex $0$. Call this ...
1
vote
1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
1
vote
2answers
75 views

Expected number of coin tosses needed until all coins show heads

We flip $n$ fair coins every iteration of the game. Every coin that shows heads is removed from the game and we use the remaining $n-k$ coins to play the game again (where $k$ is the number of heads ...
0
votes
0answers
21 views

Threshold function for component of size $k$

Show that, for each fixed $k$, there is a function $p(n)$ such that the probability that $G(n,p(n))$ has a component of size exactly $k$ tends to $1$ as $n \rightarrow \infty$. My initial thoughts are ...
6
votes
1answer
156 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
1
vote
0answers
30 views

Random 0-1 matrices [duplicate]

For $n,r∈N$, $1<r<n$, let $z(r,n)$ be the largest possible number of 0 entries in an $n×n$ matrix which has no $r×r$ submatrix whose entries are all $0$. (Here a submatrix is obtained by ...
3
votes
0answers
85 views

Random $0-1$ matrices

I'm working my way through the Oxford notes in Probabilistic Combinatorics and came across this question in one of the question sheets; I'd like to stress that this is not my homework: I'm simply ...
1
vote
0answers
25 views

Convex Hull of discrete points

If i was to give an $n \times n$ grid with each grid point having probability $p$ of being selected, would it be difficult to calculate distributions of various measures regarding the convex hull of ...
0
votes
1answer
76 views

A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

State any assumption(s) you make Well, I decided to draw a circle with a center at the origin of a Cartesian plane. It had radius r so it's coordinates on the axes were (0, r), etc. I then drew ...
0
votes
1answer
17 views

On a real line R points a,b are randomly selected such that -2<=a<=2 and 0<=b<=3. Find the probability that | a - b | > 1

Let's say that C is the set where |a-b|>1 So I suppose you could say plot it as coordinates where the x-axis (labelled a) is from [-2,2] and the y-axis (labelled b) is from [0,3]. Now |a-b| must be ...
0
votes
1answer
30 views

drawing balls from box without replacemnt

In box we have $n$ black and $m$ white balls without replacement. Let's denote $B_k$ - number of black balls drawed in first $k$ draws $W_k$ - same for white ones Let's assume that we drawed ...
2
votes
1answer
38 views

Removing a fixed quantity from multiple “buckets” randomly

Suppose I have a set of $100$ elements split into $4$ buckets A-D as follows: A: 10 elements B: 20 elements C: 30 elements D: 40 elements I want to remove $k ...
5
votes
2answers
2k views

A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads? [duplicate]

I'm studying for a probability exam and came across this question. I watched the video solution to it but I don't really understand it. I was hoping someone could explain this problem to me. Are there ...
0
votes
1answer
83 views

Help needed to derive combinatorics formula.

I am having troubles understanding a combinatorics formula. I would appreciate any ideas or hints, leading to an explanation how this formula might be derived. I came across the formula reading a book ...
0
votes
1answer
31 views

probability of $k$ boxes contain 1 ball

Occupancy problem with ball and box. Suppose there are $N$ balls and $M$ boxes. The balls are thrown to the boxes at random. What is the probability of $k$ boxes contain 1 ball? where ...
2
votes
2answers
68 views

Lower bounding the number of children in branching process

Suppose we have a recursive branching process where the number of children is given by $n p$ for parameters $n$ and (probability) $p$. Each child branches $n p$ children (once) and these children ...
2
votes
2answers
41 views

Distribution of suits in a 13 card hand

Let's say you have 13 cards distributed from a standard deck, find the probability of this distribution of suits: 4, 4, 3, 2, (for instance 4 hearts, 4 clubs, 3 diamonds, 2 spades). My answer was: ...
1
vote
1answer
42 views

Drawing without replacement yields identically distributed sequences

This question is inspired by my interest in this answer by Andre, and is related to advancing my background in combinatorics overall. How can we show formally the following fact. If we draw ...
0
votes
1answer
43 views

Expected number of $k$-cliques in $G(n, 1/2) \ge 1$

Let the expected number of $k$-cliques be denoted by $$f(k) = \binom{n}{k} (\frac{1}{2})^{- \binom{k}{2}}$$ let $k_0$ denote the largest $k$ such that $f(k) \ge 1$. I want to prove that $k_0 = ...
0
votes
0answers
51 views

Marginal and conditional probability table without joint probability table

I've a Bayesian network, with discrete node values: for every node I've the conditional probability table $p(A|B)$, where $A$ is the node itself and $B$ is the set of the parents nodes. Now I would ...
2
votes
1answer
81 views

probability of a run of 100 6s in an infinite number of rolls of a die

i came upon this problem that i couldn't solve they way i wanted. Basically, fair die is being rolled infinitely many times. Prove that the probability of there somewhere being 100 consecutive 6's is ...
0
votes
2answers
43 views

Probability of selecting one of multiple sets of distinct items

Here is the problem I am having: You have a set of items; let's say colored stones. There are 40 stones. 3 Blue, 3 Red, 3 Green, 3 White, 3 Yellow, 3 Purple, 3 Orange, 1 Black, 18 Grey. Without ...
0
votes
0answers
30 views

Probability: Disease and Diagnosis

The probability of occurrence of a certain disease in a population is $1/101$. A diagnostic test has $9$ out of $10$ chances to detect the disease when the tested subject is actually affected. On the ...
1
vote
2answers
37 views

$K$ events that are $(K-1)$-wise Independent but not Mutually/Fully Independent

I had the following question: Construct a probability space $(\Omega,P)$ and $k$ events, each with probability $\frac12$, that are $(k-1)$-wise, but not fully independent. Make the sample space as ...
0
votes
1answer
25 views

Stack of 20 cards. Distribute to 4 people randomly… What is the probability each person ends up with the same number of cards?

Stack of 20 cards. Distribute to 4 people randomly. Assume that each person does not have t be dealt the same number of card. What is the probability each person ends up with the same number of ...
1
vote
3answers
123 views

Probability of Head in coin flip when coin is flipped two times

Probability of getting a head in coin flip is 1/2. If the coin is flipped two times what is the probability of getting a head in either of those attempts? I think both the coin flips are mutually ...
0
votes
0answers
39 views

Counting problem about sub-matrices

EDIT (along the lines suggested by @sea turtles): Given a prime power $q$ and a positive integer $x\gt q$, how many subsets $A\subseteq{\bf F}_2^q$ have size $x$ and contain a subset $B\subseteq A$ ...
1
vote
1answer
37 views

An extended “birthday sharing” problem; sharing multiple properties

== THE SETUP == The table below shows the probability that Person N will draw a specific color ball out of a bag. Each person is given one opportunity to draw each color ball. For example, person 1 ...
1
vote
1answer
38 views

The Probability $P_{[m]}$ that exactly $m$ among the $N$ events $A_1,…A_n$ occur simultaneously

For $1\le i_1,i_2,...,i_k\le N$ denote $$p_{i_1,...,i_k}=\Pr(A_{i_1}\cap A_{i_2}\cap\dots\cap A_{i_k}),$$ $$S_k=\sum\limits_{1\le i_1\le\dots\le i_k\le N}p_{i_1,\dots,i_k}.$$ Show that ...
1
vote
1answer
70 views

There is 10 people that pick-up random number between 1 to 20

There is 10 people that pick-up random number between $1$ to $20$. More then one person can pick up same number (i.e. the pick-ups are independent). What is the probability that the minimum number of ...
2
votes
4answers
68 views

determining the amount of total questions needed in a game given the probabilty

I'm creating a game and can't seem to quite figure this out - driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn't change. so ...
2
votes
1answer
48 views

Probability Question I can't get around.

This is the question from my assignment, which I can't get around. Suppose that a water distribution system is composed of a number of independent pipes. At temperatures below 0 deg C, the pipes ...
1
vote
1answer
43 views

Condorcet winner and pluralism by 51%

Is it possible for the winner by pluralism to not be the Condorcer winner, if the winner by pluralism wins by 51%? This assumes 3 or more candidates. Also this assumes a rank and or preferential ...
0
votes
1answer
45 views

If $\mathbb P(A_r\cap A_s)=p$ and $\mathbb P(A_r\cap A_s\cap A_t)=q$, express $q$ in terms of $p$

Let $A_r$, ($1\le r\le n$) be events such that, $\mathbb P(A_r\cap A_s)=p$ and $\mathbb P(A_r\cap A_s\cap A_t)=q$, ($r<s<t$). If the probability of at least two occurring is $\dfrac{1}{2}$ ...
0
votes
0answers
33 views

Threshold function for non-balanced graphs counter example

Let $G(n,p)$ be a random graph and let $H$ be a balanced graph with $e$ edges and $v$ vertices.. We know that $p^{*}(n)=n^{-v/e}$ is the threshold function for containing a copy of $H$. That is, if ...
0
votes
0answers
20 views

$\exists A\in E,\forall\epsilon>0,\exists I_{\epsilon}\in P_f(I),\forall J\in P_f(I),I_{\epsilon}\subset J\Rightarrow ||A-A_J||\le\epsilon$

Can you explain me the following definition please. I understand what the statement in general means, but I think there are some typos Let $E$ be a real or complex vector space with norm ...
2
votes
0answers
44 views

We are giving $m$ prizes to $n$ people at lottery…

We are giving $m$ prizes to $n$ people at lottery... Question A: What is the probability that no one will get more then one prize (assume that $n\ge m$). Question B: What is the probability the ...
1
vote
1answer
39 views

Show that $(u_1+u_1+…u_k)^n=\sum\limits_{r_1+r_2+…r_k=n}\dfrac{n!}{r_1!r_2!…r_k!}u_1^{r_1}u_2^{r_2}…u_k^{r_k}$

Let $r_1,...r_k$ be integers sucht that, $r_1+r_2+r_3+...,r_k=n$ The number of ways in which a subpopulation of $n$ elements can be partitioned into $k$ subpopulations of which the first ...
2
votes
2answers
137 views

What does $E[XY]$ mean?

Let's say I have two random variables, $X$ and $Y$. $X$ is the value of a fair die, $Y$ is the result of a coin flip, with heads being 1 and tails being 0. $E[X] = \sum_{k=1}^{6}{\frac{k}{6}} = ...
1
vote
0answers
86 views

How to calculate the probability that the average of a multinomial process exceeds some value

I've been mulling over a problem that has something like the following form. I don't have a math or stats background so advice and answers at various levels, from terminological to strategic, would be ...
1
vote
0answers
93 views

Randomized packing of items

An adversary gives you a set of items whose total size is $x$ (he gets to choose how $x$ is distributed. e.g. there may be $k-1$ items of size $\frac{x}{k}$ and 2 items of size $\frac{x}{2k}$). The ...
14
votes
1answer
236 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemple. How to calculate the number of circuits that visit ...
0
votes
1answer
59 views

Find $E[XY]$ of joint hypergeometric by conditioning on $Y$

An urn contains $N = 30$ balls. There are $10$ balls of color X, and $8$ balls of color Y, and let random variables $X, Y$ denote the count of each, respectively. Also assume $n = 12$ balls are ...
0
votes
1answer
255 views

What is the probability of picking Exactly 1 red marble and than not 1 red marble? without rep.

A urn has 3 red marbles, 2 blue marbles, 1white, 1 black 1 brown. What is the probability of getting exactly 1 red marble than not 1 red marble? What is the probability of getting at least 1 red ...
-3
votes
1answer
33 views

combinatorics -probability of burned electric bulbs in row

Given 50 bulbs in a row.In a given time (not maintenance problem) the probability of a burned bulb is 0.1 . Please calculate the probability that the last 5 bulbs in row are burned. I really doubt ...
1
vote
1answer
56 views

Number of ways to form three distinctive items

Given a 6 by 5 array, Calculate the number of ways to form a set of three distinct items such that no two of the selected items are in the same row or same column. What I did was $C(30,1) \cdot ...