Pascal's Traingle

Pascal's triangle is an arrangement of the coefficients from a binomial distribution placed into a triangle shape. In most English speaking countries it is named after Blaise Pascal, but the triangle appears in many other countries like India, Persia, China, and Italy where it was studied by mathematicians centuries before Pascal began his work.

4 rows of pascal's TriangleThe rows of Pascal's triangle represent the number of answers you get when you do something that is a binomial distribution. To simplify, we will think of tossing coins, where the only possibility is that you get either a head or a tail.

The very top row - 1 - is the answer you get if you toss NO coins. This is kind of weird, but the one answer you get is "nothing happened". Remember - serious math people don't always think like most people, so you might have to trust me on this one. Statisticians and mathematicians call this top row "row zero" and if you take one of my math classes you will learn why.

On the second row - 1   1 - is the answer you get if you toss ONE coin. You get either a head or a tail. Some people would say the chances are 50 - 50, meaning you get a head about half the time and a tail the other half.

Now it starts to get a little more interesting. On the third row down - 1   2   1 - is the answer you get if you toss TWO coins. Notice that the number of coins is 2, the number after the very first 1 in the row. For coin examples, this is always true. On this row we find out that we have one way to get both heads, 2 ways to get a head and a tail, and only one way to get both tails. If you add up the numbers 1 + 2 + 1 it equals 4, so you have 1 chance in 4 of getting both heads. 1 divided by 4 is .25 or 25%. Now you can see that you don't actually need to toss coins thousands of times to figure out the probabilities, you only need to go to the correct row of Pascal's Triangle

Lets jump down to the bottom row in the triangle image above. This row says: 1    4    6    4    1.

Since the second number in the row is a 4 then this must be the row for tossing 4 coins at the same time. if we add up the numbers in the row, we get 1 = 4 + 6 + 4 + 1 = 16. Combining that with the very first 1, we know that there is only one way out of 16 to get 4 heads when you toss your coins. If you take 1 divided by 16 you get .0625 or 6.25%.

Sierpinski TriangleBesides using Pascal's Triangle for probability, it is also used in algebra for binomial expansions - things like (x + y) to the n power. I will not explain that here, but it comes in very handy in Algebra 2 and calculus.

There are other patterns and properties related to Pascal's Triangle. If you draw patterns of little triangles, instead of the numbers in the triangle at the top of the page, and then color in only the odd numbers you get a pattern that looks like the fractal called a Sierpinski triangle. The more rows of Pascal's Triangle that you use, the closer the pattern gets to an accurate Sierpinski triangle. If you had a lot of time and could do this for an infinite number of rows it is exactly the Sierpinski triangle.

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Last Updated: 8/10/2007
K. A. Shearer 2007
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