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Undergraduate
By
Shiva Kintali
on May 21, 2014 | Updated Jan. 4, 2018
99 fair coins
Person \(A\) flips 99 fair coins and obtains \(a\) heads. Person \(B\) flips 100 fair coins and obtains \(b\) heads. What is the probability that \(a < b\) ?
Mathematics
Probability
conditional probability
interview question
0
Graduate
By
Shiva Kintali
on Aug. 29, 2012 | Updated Dec. 6, 2017
Random Intervals
There are \(n\) points on a line. These points are paired up at random to form \(n/2\) intervals. Prove that the probability that among these intervals there is one which intersects all the others i…
Mathematics
Puzzles
Probability
Puzzles
counting
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Undergraduate
By
Shiva Kintali
on June 17, 2013 | Updated Dec. 6, 2017
Random cars
The probability of observing a car during a 30 minute period on a road is 95%. What is the probability of observing a car in a 10 minute period ?
Mathematics
Puzzles
Probability
Puzzles
interview question
probability
0
Undergraduate
By
Shiva Kintali
on June 10, 2012 | Updated Dec. 6, 2017
Stick triangle
A stick is broken at random into three pieces. What is the probability that the pieces can form a triangle?
Mathematics
Puzzles
Geometry
Probability
Puzzles
math puzzle
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Murphy's Law
Let \(A_1, A_2, \ldots ,A_n\) be independent events, and let \(T\) be the number of these events that occur. Show that the probability that none of the events occur is at most \(e^{-E[T]}\), wh…
Mathematics
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on Nov. 14, 2012 | Updated Dec. 6, 2017
Randomized k-SAT algorithm
Consider an instance of SAT with \(m\) clauses, where every clause has exactly \(k\) literals. Give a Las Vegas algorithm that finds an assignment satisfying at least \(m(1-2^{-k})\) clauses, and an…
Mathematics
Probability
conditional expectation
derandomization
expectation
sat
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Runs in coin tosses
You have a biased coin i.e., each coin toss is a head with probability \(p\) and is a tail with probability \(1-p\). Let \(X\) be the number of runs in \(n\) independent tosses of this coin. Here, run…
Mathematics
Probability
expectation
variance
0
Undergraduate
By
Shiva Kintali
on June 6, 2013 | Updated Dec. 6, 2017
Two points
Two points are chosen randomly and independently from the interval \([0,1]\) according to uniform distribution. What is the expected distance between the two points ?
Mathematics
Probability
expectation
interview question
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Randomly built binary search tree
Define a randomly built binary search tree on \(n\) keys as one that arises from inserting the keys in random order into an initially empty tree, where each of the \(n!\) permutations of the input key…
Computer Science
Mathematics
Algorithms
Data Structures
Probability
binary tree
expectation
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Basics of Random Graphs
Let \(G = G(n, \frac{1}{2})\) be a random graph on \(n\) vertices, i.e., for each pair of verices \(i, j\), we add the edge \((i, j)\) independently with probability \(\frac{1}{2}\). Show that …
Mathematics
Probability
high probability
random graphs
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