Q 235. The Chinese “Seven multiplying squares” became the Pascal’s Triangle several centuries later and then gave rise to binomial theorem. Explain how these relate to some of the basics of probability. 

 

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

• All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/
• Please visit the forum home page at https://www.benchmarksixsigma.com/forum/ to respond to the latest question open till the next Tuesday/ Friday evening 5 PM as per Indian Standard Time
• The best answer is always shown at the top among responses and the author finds honorable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term

The Pascal's triangle is an imaginary triangle in mathematics, that was discovered way back and used by mathematicians of different  countries like India , Iran, China, Germany, Italy  and finally highlighted by French mathematician Blaise Pascal, who is encredited with this phenomenon. It is applicable for binomial distributions and contains binomial coefficients, arranged in triangular array. It finds probability of events  and combination of events.  The sum of  numbers in rows in Pascal triangle is given by 2ⁿ. Any probability evaluation, with two equally, independent and no predetermined order can be resolved , using Pascal’s triangle. 

The initial  row of Pascal's triangle is  conventionally designated as  the 0 th row, n=0 at the top. The value of  0 th row is assigned as a  non zero  value and usually assigned as 1.  The entries in each row are  numbered from left  and both extreme ends of a row are assigned values of 1.  Each  value entered in next row is the sum of value  in the above  and  to the left with above and to the right.

 

Example : 

A group of 10 people  needs to be picked  to create a committee of 4 people. We need to figure out the number of possible different committees of size 4 , that can be created from 10 people. While solving this issue, combination of people is important, not the mentioned order of the people. There will  ₁₀C₄ possible committees. By scrutinizing, 10th row of Pascal's Triangle and selecting  over to the 5th term (As first term is ₁₀C₀),  it  will give us the number of possible different committees. 
 
So we can conclude that there will be  210 possible committees of 4 people each, from a group of of 10 people.

 

 

 

 

 

 

Applications of Pascal’s Triangle:

 

-       Algebra and probability

-       Graphic designers

-       Finance

-       Architect

-       Mapping
 

 

Benchmark Six Sigma Expert View by Venugopal R

Pascal's Triangle is named after the French Mathematician Blaise Pascal. It will look as depicted below:

 

 

A quick examination about the Pascal’s triangle reveals the following:

 

The top most row (referred to as 0th row) has one number, which is 1. The next row (first row) has two numbers (or two columns) and each number is the sum of the numbers of the boxes above from the previous row. The same practice continues, and we get the Pascal’s triangle. Thus if number on the n^th row and k^th column is represented as  then:

Let us look at an example of a simple binomial probability – the outcome of tossing a coin. The following table gives the number of tosses, the outcome and the numerical representation of each outcome combination

 

 

The last column of the above table is emerging as the Pascal’s triangle. It may also be seen that the binomial probabilities for a particular outcome can be worked out. For example, let’s see the probability of obtaining exactly two heads, when the coin is tossed 4 times.

 

The total number of possible outcomes is 1+4+6+4+1 = 16. The number of combinations that gave exactly two heads is 6. Hence the probability of obtaining exactly two heads is 6 /16 = 0.375 or 37.5%

 

 

 

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

To build the triangle, start with "1" at the top, then continue the number series below 1 in a triangular pattern.

Each number placed in the triangle, is the sum of the numbers above it.

Pattern of the Triangle

The first diagonal is, of course, just "1"s

The next diagonal has the counting numbers (sum of numbers above) (1,2,3, etc).

The third diagonal has the triangular numbers

The fourth diagonal has the tetrahedral numbers

ODDS & EVENS

If the odds and evens are separated using different colours then the pattern ends up in Sierpinski triangle.(ever repeating pattern of triangles)

Pascal’s Triangle & Probability

Pascal’s triangle can be used to find the probability of any combinations

The number of combinations of Head’s & Tails when a coin is tossed can be identified using Pascal’s Triangle. Thus helps us in identifying the probability.

For example, if we toss a coin 2 times, there is only one combination that will give you 2 heads  or 2 tails (HH), but there are two combinations that will give 1head and one tail (HT,TH), This is the pattern "1,2,1" in Pascal's Triangle.

Pascal's triangle is made up of the coefficients of the Binomial Theorem which is the sum of a row n is equal to 2n. any probability problem that has two equally possible outcomes can be solved using Pascal's Triangle.

Very good answers by T Jayaram and Shashikant Adlakha. Both are winners. 
 

Also, do review the response from Mr Venugopal for expert view.