# Tagged Questions

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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 } ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $K$ 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 ...
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### 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 ...
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### 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 ...
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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 $B\subseteq A$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\dfrac{1}{2}$ ...
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### 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 ...
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### $\exists A\in E,\forall\epsilon>0,\exists I_{\epsilon}\in P_f(I),\forall J\in P_f(I),I_{\epsilon}\subset J\Rightarrow ||A-A_J||\le\epsilon$

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 ...
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### 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 ...
### 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 ...