We consider a simple problem in probability: A thin rod is broken at random into three pieces. What is the probability that these three pieces can be used to form a triangle? This problem is solved without difficulty. If the rod is of unit length, aligned along the interval $latex {0\le x \le 1}&fg=000000$, we … Continue reading Breaking a Stick to Form a Triangle
Tag: Probability
Limits of Sequences, Limits of Sets
In undergraduate mathematics, we are confronted at an early stage with "Epsilon-Delta" definitions. For example, given a function $latex {f(x)}&fg=000000$ of a real variable, we may ask what is the value of the function for a particular value $latex {x=a}&fg=000000$. Maybe this is an easy question or maybe it is not. The epsilon-delta concept can … Continue reading Limits of Sequences, Limits of Sets
Benford’s Law Revisited
Several researchers have observed that, in a wide variety of collections of numerical data, the leading --- or most significant --- decimal digits are not uniformly distributed, but conform to a logarithmic distribution. Of the nine possible values, $latex {D_1=1}&fg=000000$ occurs more than $latex {30\%}&fg=000000$ of the time while $latex {D_1=9}&fg=000000$ is found in less … Continue reading Benford’s Law Revisited
Bernoulli’s Golden Theorem and the Law of Large Numbers
Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli's great work, Ars Conjectandi, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, … Continue reading Bernoulli’s Golden Theorem and the Law of Large Numbers
Think of a Number: What are the Odds that it is Even?
Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime? Since the set $latex {\mathbb{N}}&fg=000000$ of natural numbers is infinite, there are difficulties in assigning probabilities to subsets of $latex {\mathbb{N}}&fg=000000$. We require the probability … Continue reading Think of a Number: What are the Odds that it is Even?
How many numbers begin with a 1? More than 30%!
The irregular distribution of the first digits of numbers in data-bases provides a valuable tool for fraud detection. A remarkable rule that applies to many datasets was accidentally discovered by an American physicist, Frank Benford, who described his discovery in a 1938 paper, "The Law of Anomalous Numbers" [TM181 or search for “thatsmaths” at irishtimes.com]. … Continue reading How many numbers begin with a 1? More than 30%!
The Two Envelopes Fallacy
During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed … Continue reading The Two Envelopes Fallacy
The Improbability Principle
Extremely improbable events are commonplace. “It's an unusual day if nothing unusual happens”. This aphorism encapsulates a characteristic pattern of events called the Improbability Principle. Popularised by statistician Sir David Hand, emeritus professor at Imperial College London, it codifies the paradoxical idea that extremely improbable events happen frequently. [TM112 or search for “thatsmaths” at irishtimes.com]. We … Continue reading The Improbability Principle
Treize: A Card-Matching Puzzle
Probability theory is full of surprises. Possibly the best-known paradoxical results are the Monty Hall Problem and the two-envelope problem, but there are many others. Here we consider a simple problem using playing cards, first analysed by Pierre Raymond de Montmort (1678--1719). Montmort's Problem Take two piles of cards faced down, one with the 13 … Continue reading Treize: A Card-Matching Puzzle
Twenty Heads in Succession: How Long will we Wait?
If three flips of a coin produce three heads, there is no surprise. But if 20 successive heads show up, you should be suspicious: the chances of this are less than one in a a million, so it is more likely than not that the coin is unbalanced. Waiting Time for a Single Head On average, … Continue reading Twenty Heads in Succession: How Long will we Wait?
Random Harmonic Series
We consider the convergence of the random harmonic series $latex \displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n} &fg=000000$ where $latex {\sigma_n\in\{-1,+1\}}&fg=000000$ is chosen randomly with probability $latex {1/2}&fg=000000$ of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is ``almost surely'' convergent. We are all familiar with the harmonic series … Continue reading Random Harmonic Series
Bertrand’s Chord Problem
The history of probability theory has been influenced strongly by paradoxes, results that seem to defy intuition. Many of these have been reviewed in a recent book by Prakash Gorroochurn [2012]. We will have a look at Bertrand's Paradox (1889), a simple result in geometric probability. Let's start with an equilateral triangle and add an … Continue reading Bertrand’s Chord Problem
Franc-carreau or Fair-square
Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player's coin lands completely within a square, he or she wins a coin of equal value. If the … Continue reading Franc-carreau or Fair-square
Twin Peaks Entropy
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks. Classical Entropy Entropy was introduced in classical thermodynamics about 150 years ago and, somewhat … Continue reading Twin Peaks Entropy
Buffon was no Buffoon
The Buffon Needle method of estimating $latex {\pi}&fg=000000$ is hopelessly inefficient. With one million throws of the needle we might expect to get an approximation accurate to about three digits. The idea is more of philosophical than of practical interest. Buffon never envisaged it as a means of computing $latex {\pi}&fg=000000$. Buffon and his Sticks … Continue reading Buffon was no Buffoon
Bent Coins: What are the Odds?
If we toss a `fair' coin, one for which heads and tails are equally likely, a large number of times, we expect approximately equal numbers of heads and tails. But what is `approximate' here? How large a deviation from equal values might raise suspicion that the coin is biased? Surely, 12 heads and 8 tails … Continue reading Bent Coins: What are the Odds?
The Faraday of Statistics
This week, That’s Maths in The Irish Times ( TM044 ) is about the originator of Students t-distribution. In October 2012 a plaque was unveiled at St Patrick's National School, Blackrock, to commemorate William Sealy Gosset, who had lived nearby for 22 years. Sir Ronald Fisher, a giant among statisticians, called Gosset “The Faraday of … Continue reading The Faraday of Statistics
Breaking Weather Records
In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each … Continue reading Breaking Weather Records
Bayes Rules OK
This week, That's Maths ( TM018 ) deals with the "war" between Bayesians and frequentists, a long-running conflict that has now subsided. It is 250 years since the presentation of Bayes' results to the Royal Society in 1763. The column below was inspired by a book, The Theory that would not Die, by Sharon Bertsch … Continue reading Bayes Rules OK
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