Category Archives: probability

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

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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

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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 balls-and-jars … Continue reading

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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

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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

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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

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Birth-and-Death Process Steady-State Probabilities

A birth-and-death process is a special kind of continuous-time Markov chain that has applications in queuing.  If we assume that arrivals to a queuing system follow a Poisson process and that service times are exponentially distributed, then the resulting queuing system … Continue reading

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