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EASIER THAN YOU THINK...

Probability

Tossed Coin About to Land in a Human HandWhat are the chances?

Chance is involved in so much of our lives and in so much of the physical world that it cannot be ignored.  Despite this, it was only in the mid 1600s that mathematicians really began to study chance … and the discipline of probability was born!

A French writer, and amateur mathematician, Antoine Gombaud became interested in a number of gambling problems.  One, in particular, was already centuries old.  If two players agreed to play a fixed number of games (a best of seven, for example), how should they divide the stake if the game is interrupted?  How would their existing wins (four games to one, for example) affect their prospects for ultimately winning the set of games and, therefore, how the stake should be divided?

He enlisted the help of Mersenne’s circle of educated friends and two mathematicians in particular took a particular interest in these problems.  Blaise Pascal and Pierre de Fermat corresponded with Gombaud and the two of them established the general  principles of probability theory.  Gerolamo Cardano was also involved in the discussions.

Within a couple of years, the famous Dutch scientist Christiaan Huygens (who taught Gottfried Leibniz analytical geometry and discussed Iceland Spar and other matters with Isaac Newton) had learned of this correspondence and published the first book containing a study of probability theory, De Ratiociniis in Ludo Aleae (1657).

Very quickly, probability became a popular field of study.

The Swiss mathematician, Jacob Bernoulli, wrote a seminal treatise on combinatorics and probability called Ars Conjectandi (The Art of Conjecturing).  Sadly (for him), it was published eight years after his death by his nephew, Nicolaus Bernoulli, in 1713.

The French mathematician, Abraham de Moivre, published the first complete probability text book, The Doctrine of Chances, in 1718.  Because he lived most of his life in England (as a Hugenot exile from France), this book was written in English and you may download and read a copy of it at the archive.org site.

In 1812 Pierre-Simon Laplace introduced a host of new ideas and mathematical techniques in his book, Théorie Analytique des Probabilités (you can read an English translation of the relevant section of this book thanks to the kind folk at Xavier University in Cincinnati, Ohio).  It was Laplace who first seriously applied principles of probability to scientific and practical matters.

Magnetic Dice Trick (aka Two 'Magic' Dice)

Many years ago I read a story (probably fictitious) about two boys who earned a modest income by selling pairs of dice to gamblers.

The story related that they would get the gamblers to roll two dice until they rolled 11 (a 5 and a 6) in order to 'magnetise' or 'align' the dice. Amazingly, once this had happened, the dice rolled in a particular way with very high probability!

In this video I explain and demonstrate the process ... and then analyse it. It is a wonderful exercise in probability and I hope you enjoy it. You will learn some good mathematics, and may even be a little more wary of young boys offering you something that looks too good to be true!

By way of explanation, I like to think and talk of three kinds of probability. There is a theoretical probability (the kind that we often calculate in school when analysing a particular situation). There is experimental probability, where our estimates are based on past experiences (this is the basis for our weather forecasts, for example). And I like to talk of 'perceived' or 'psychological' probability (which is the intuitive way that most people seem to think of a problem). The world is full of examples of people who wrongly estimate their real chances of winning something or succeeding at something. This is because there is a big difference between their personal 'perceived' chances and their actual (theoretical or experimental) chances. The Darwin Awards, for example, record the results of such miscalculations! The example I discuss in this video is effective because it is based on exactly this sort of disparity ... between our perceptions and reality.

Enjoy!

How I wish you could be my professor – I have been having a consistent problem with the chain rule for what seems like weeks. I couldn’t grasp it at all. The first couple minutes of your video – it all instantly clicked for me. You are a life saver and a wonderful teacher. Thank you so much for your explanations!!!!!!!
Taikrys (on a CCM YouTube video about the Chain Rule)

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