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Category Archives: probability
A CoinFlipping Problem
One problem that I’ve assigned when discussing Markov chains is to calculate the expected number of flips required for a particular pattern to appear. (Here I mean a pattern such as heads, heads, heads, or HHH.) In this post I’m … Continue reading
Posted in coin flipping, Markov chains, probability
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Equivalence of Two Binomial Sums
This month’s post entails proving the following equivalence: Identity 1: Despite the fact that these two sums are over the same range and are equal for all values of n they are not equal termbyterm. If you think about the two sides … Continue reading
Posted in binomial coefficients, probability
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The Secretary Problem With the Two Best
The secretary problem is the following: Suppose a manager wants to hire the best person for his secretary out of a group of n candidates. He interviews the candidates one by one. After interviewing a particular candidate he must either (1) … Continue reading
Posted in optimization, probability
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Counting Poker Hands
For this post I’m going to go through a classic exercise in combinatorics and probability; namely, proving that the standard ranking of poker hands is correct. First, here are the standard poker hands, in ranked order. Straight flush: Five cards of … Continue reading
Posted in combinatorics, probability
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An Expected Value Connection Between Order Statistics from a Discrete and a Continuous Distribution
Years ago, in the course of doing some research on another topic, I ran across the following result relating the expected values of the order statistics from a discrete and a continuous distribution. I found it rather surprising. Theorem: Fix n, and … Continue reading
Posted in order statistics, probability, statistics
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A Probabilistic Proof of a Binomial Coefficient Identity, Generalized
In a post from a couple of years ago I gave a probabilistic proof of the binomial coefficient identity In this post I modify and generalize this proof to establish the identity As in the original proof, we use a ballsandjars … Continue reading
Posted in binomial coefficients, probability
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Independence of the Range and Minimum of a Sample from an Exponential Distribution
A few years ago I answered a question on math.SE about the distribution of the sample range from an exponential (1) distribution. In my answer I claim that the range and the minimum of the sample are independent, thanks to the … Continue reading
Posted in probability, statistics
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Picking Random Points on a Circle (Not a Disk)
How do you select points at random on a circle? By “circle” I mean the outside of a disk, not its interior. In this post I’m going to discuss two methods: (1) Selecting an angle at random on and taking … Continue reading
Posted in probability, simulation, statistics
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A Probabilistic Proof of a Binomial Coefficient Identity
I’m a big fan of combinatorial proofs, in which you show that both sides of an identity count the same entity in two different ways. Probabilistic proofs – in which you show that both sides of an identity express the probability … Continue reading
Posted in binomial coefficients, probability
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