A toss of two dice may throw two sixes but it is also possible that it may throw a combination of two other numbers. How many different bracelets can be made by stringing ten pearls, seven rubies, six emeralds and eight sapphires if similar stones are to be kept apart? Answers to such questions demand a laying out of all possible cases but often they are too many to be counted. In such a case, we satisfy ourselves with just having a method to enumerate all possible cases without actually counting them. This is probability. Theory of probabilities is at bottom only common sense reduced to calculation. It is remarkable that this theory has become the most important object of human knowledge.
The founders of the theory of probability were Blaise Pascal and Fermat. This theory was developed in an interesting exchange of letters between them during the year 1654. The correspondences contain methods for tedious enumeration of all possible cases. Pascal suggested a short cut but fell into error and was corrected by Fermat. They began discussing this theory when their friend, Chevalier de Mere, a professional gambler, wanted to know his chances of winning at different stages of any game and sought help from his mathematician friends. Pascal made extensive use of delightful arithmetical triangle in giving his solutions. Arithmetical triangle is given below:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...................................................................................................................................................................,
In this triangle, the numbers in any row after the first two are obtained from those in the preceding row by copying the terminal 1’s and adding together the successive pairs of numbers from left to right to give the new row.
Blaise Pascal is best known to us apart from his theory of probability, for his invention of calculator and his two literary classics Pensees and Provincial Letters. It is little known that he suffered all his life from bad stomach and strange religious inclination of tormenting Jesuits. He suffered from them so badly that it is said that the great gifts were bestowed upon a wrong person. His father banned him from studying mathematics fearing it may strain his brains and therefore his health. He once curiously asked his father about what geometry is all about. The description given by his father launched his interest in mathematics and gave the world, the greatest might-have-been in mathematical history. Legend goes that he intuitively proved several of Euclid’s propositions before he actually saw Euclid’s book “Elements”. His first spectacular achievement was to prove that sum of the angles of a triangle is 180 degrees. He is also credited with some of the most beautiful discoveries in mathematics like Cat’s Cradle and Mystic Hexagram.
After their father’s death, he came to Paris with Jacqueline his younger sister. She joined nunnery at Port Royal and converted her brother to Jansenism thus burying his talent forever. In next eight years, he turned his back on the world and quit pursuits of the flesh and the mind. Only once did he fall from grace before his death and that was when he solved the mystery of Cycloid, which is called the “Helen of Geometry” as he contemplated the mystery, as his tooth was being extracted and he was undergoing excruciating pain.
I am sorry, the arithmetical triangle is not appearing as a triangle.
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