Probability in Ancient India

A first course on probability (at the high-school level) typically begins with an account of the theory of permutations and combinations needed for calculating probabilities in games of chance, such as dice or cards. Like many other aspects of mathematics, this theory of permutations and combinations first developed in India, although an account of its history is usually missing in stock presentations of combinatorics. In fact, the theory of permutations and combinations was basic to the Indian understanding of metre and music. The Vedic and post-Vedic composers depended on combinations of two syllables called guru (deep, long) and laghu (short). The earliest written account of this theory of metre is in Piñgala’s Chandahsūtra (−3rd c. CE), a book of aphorisms (sūtra-s) on the theory of metre (chanda). To calculate all possible combinations of these two syllables in a metre containing n syllables, Piñgala gives the following rule (which explicitly makes use of the symbol for zero). “(Place) two when halved;” “when unity is subtracted then (place) zero;” “multiply by two when zero;” “square when halved.” In a worked example, Dutta and Singh show how for the Gāyatr̄ı metre with 6 syllables this rule leads to the correct figure of 2 possibilities. That this rule basically involves the binomial expansion is made clear by Piñgala’s commentator the 10 c. CE Halayudha. Thus, in a 3-syllabic metre with two underlying syllables, guru and laghu, 3 guru sounds will occur once, 2 gurus and 1 laghu will occur twice, as will 1 guru and 2 laghus, while 3 laghus will occur once. Symbolically (g + l) = g + 3gl + 3gl + l. To generalize this to the case of n underlying syllables, Halayudha explains the meru-prastāra (pyramidal expansion) scheme for calculation, which is identical to “Pascal’s” triangle which first appeared in Europe about a century before Pascal (on the title page of the