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Undergraduate
By
Shiva Kintali
on Nov. 12, 2012 | Updated Dec. 6, 2017
Alice, Bob and children
Alice and Bob decide to have children until either they have their first girl or they have \(k \geq 1\) children. Assume that each child is a boy or girl independently with probability 1/2, and that …
Mathematics
Probability
expectation
0
Undergraduate
By
True Putnam
on Aug. 28, 2013 | Updated Jan. 4, 2018
Putnam 2005 A6
Let \(n\) be given, \(n \geq 4\), and suppose that \(P_1,P_2, \dots,P_n\) are \(n\) randomly, independently and uniformly, chosen points on a circle. Consider the convex \(n\)-gon whose vertices are \…
Mathematics
Probability
uniform distribution
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Minimum of Random Subsets
Let \(S = \){\(1,2,\dots,n\)}. Let \(A,B\) be two random subsets of \(S\). Let \(\min(A)\) denote the minimum number in the set \(A\). What is the probability that \(\min(A)= \min(B)\) ? Evalua…
Mathematics
Probability
probability
0
Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Dec. 6, 2017
Balls and bin game
Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following : choose one ball from the bin uniformly at random, and then put the…
Mathematics
Probability
sampling
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Detecting fair coin
You are given two coins: one is a fair coin and the other is a biased coin that produces heads with probability 3/4. One of the two coins is picked, and is tossed \(n\) times. How many tosses suffic…
Mathematics
Probability
conditional probability
0
Undergraduate
By
Shiva Kintali
on Oct. 23, 2012 | Updated Dec. 6, 2017
Randomly assigned jobs
Suppose we have \(n\) jobs to distribute among \(m\) processors. For simplicity we assume that \(m\) divides \(n\). A job takes one step with probability \(p\) and \(k > 1\) steps with probability …
Mathematics
Probability
chernoff bound
0
Undergraduate
By
Shiva Kintali
on Oct. 3, 2013 | Updated Jan. 4, 2018
k-partite subgraph
For each \(k\in \mathbb{N}\) and each simple graph \(G=(V,E)\), prove that \(G\) has a \(k\)-partite subgraph \(H=(V',E')\) (i.e., \(H\) has chromatic number at most \(k\)) such that …
Mathematics
Graph Theory
Probability
graph coloring
probabilistic method
0
Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Dec. 6, 2017
Basics of probability
Prove that, if \(E_1, E_2, \dots, E_n\) are mutually independent, then so are \(\overline{E_1}, \overline{E_2}, \dots, \overline{E_n}\) Give an example of three random events \(X,Y,Z\) for which any …
Mathematics
Probability
basics
conditional probability
0
Undergraduate
By
Shiva Kintali
on Oct. 21, 2012 | Updated Dec. 6, 2017
Number of inversions
Let \(A\) be an array of \(n\) distinct numbers. If \(i < j\) and \(A[i] > A[j]\), then the pair \((i,j)\) is called an inversion of \(A\). Design an algorithm that determines the number of inversio…
Mathematics
Probability
expectation
0
High School
By
True IITJEE
on Jan. 9, 2017 | Updated Jan. 3, 2018
JEE Advanced 2016 Paper 1 Mathematics Question 40
A computer producing factory has only two plants \(T_1\) and \(T_2\). Plant \(T_1\) produces 20% and plant \(T_2\) produces 80% of the total computers produced. 7% of computers produced in the factory…
Mathematics
Probability
conditional probability
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