Background
Probability Theory
Pascal's Triangle & Probability
Application of Probability Theory
Pascal's wager
Objections
Homework
Joyce Lam Nga Ching
2001714828
Phil1007
12-4-2002
27-4-2002
Probability
Theory
Pascal's triangle-Introduction
Pascal's Triangle - useful tool in probability theory.
The Pascal
triangle (actually known long before Pascal) is a table of the binomial
coefficients where the (n, k)th entry is 
.
Each entry is the sum of the pair immediately above it.
Pascal publishes Treatise on the Arithmetical Triangle on "Pascal's triangle". It had been studied by many earlier mathematicians. He employed his arithmetical triangle in 1653, but no account of his method was printed till 1665. The triangle is constructed as in the figure below, each horizontal line being formed form the one above it by making every number in it equal to the sum of those above and to the left of it in the row immediately above it; ex. gr. the fourth number in the fourth line, namely, 20, is equal to 1 + 3 + 6 + 10.
It is clear from the above table that any number in the above table is equal to the sum of the numbers above it and on its left - with the exception of the first row and the first column where every number is one.
The numbers in each line are what are now called figurate numbers. Those in the first line are called numbers of the first order; those in the second line, natural numbers or numbers of the second order; those in the third line, numbers of the third order, and so on. It is easily shewn that the mth number in the nth row is (m+n-2)! / (m-1)!(n-1)!
Pascal's arithmetical triangle, to any required order, is got by drawing a diagonal downwards from right to left as in the figure. The numbers in any diagonal give the coefficients of the expansion of a binomial; for example, the figures in the fifth diagonal, namely 1, 4, 6, 4, 1, are the coefficients of the expansion (a+b)4 . Pascal used the triangle partly for this purpose, and partly to find the numbers of combinations of m things taken n at a time, which he stated, correctly, to be (n+1)(n+2)(n+3) ... m / (m-n)!
Pascal's triangle is an arithmetical triangle made up of staggered rows of numbers:
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Row 4 | |
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Row 5 | |
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A triangle of numbers arranged in staggered rows such that
 where 
is a binomial coefficient. The triangle was studied by B. Pascal |
Constructing Pascal's Triangle
Start with the top two rows, which are 1 and 1 1. To find any number in the next row, add the two numbers above it(i.e. the one above it and to the right, and the one above it and to the left). At the beginning and the end of each row, when there's only one number above, put a 1. You might even think of this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add up the number above and to the left (since there is no number there, pretend it's zero) and the number above and to the right (1), and get a sum of 1.
When people talk about an entry in Pascal's Triangle, they usually give a row number and a place in that row, beginning with row 0 and place 0. For instance, the number 20 appears in row 6, place 3.
Different Number Patterns--Finding
Numbers in Pascal's Triangle
Go to Application on Probability Theory
Reference:
1.http://mathforum.org/workshops/usi/pascal/pascal_intro.html
2.http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html