Updated September 2019

**Section Goals:**

- Develop capability to build 'Pascal's Triangle'
- Understand how to write the Binomial Distribution as an equation
- Use terms from the Binomial Equation to solve real world problems
- Build a Binomial PDF graph from the Binomial Equation

**Binomial Distribution Use Criteria**

the Binomial Distribution can be used when all 4 conditions below are met:

- Results of each trial (coin flips, shell hits, failed students etc.) are independent. (i.e. if four coins are flipped, a coin landing 'heads' has no effect on the behavior of its neighbors)
- The probability of 'success' is known. (for a coin flip, it would be 50%)
- The total number of trials (e.g. the number of coins pitched in the air) is known.
- The total number of trials (e.g. the number of coins, cannon shells, students etc.) is less than 13. Optional: Why limit it to 13 Terms?

The method of solving real world problems using the Binomial Distribution will be taught by working an example. Commentary will be provided to explain different aspects of the example.

** Example Problem Statement:** Four coins are thrown in the air, and land as they choose. Develop a Binomial Equation that represents each term of the PDF.

**Step 1:** Build Pascal's Triangle with a row that has one more number, than the problem has sample items or trials (be they coins, cannon shells or kittens). Pascal's Triangle will be used to provide numbers used in the probability formulas (4 coins->5 numbers)

Ch 3 Figure 1

The Steps for creating Pascal's Triangle are almost intuitive. Simply:

- Write down the three 1's at the top of the triangle
- Every place you find a pair of numbers side by side, total them & write the answers in the row below
- Add 1's on the outside to enable creating another row. Apply Step 2 again until your triangle has enough numbers to meet the need of your number of trials. (How Many? If the number of coins, shells or whatever is n, then n+1 numbers are needed.)

Ch 3 Figure 2

Ch 3 Figure 3

Ch 3 Figure 4

Ch 3 Figure 5

Ch 3 Figure 6

The equation you just wrote, is a math representation of the the following BINOMIAL PDF. We discuss that next.

Ch 3 Figure 7

You will need a scientific calculator with a y^x function button on it. Any calculator will do, but that y^x button makes the computations easier and reduces errors.

Ch 3 Figure 7 above is the PDF for the example problem being worked. Let us verify the second probability bar (i.e. probability of 3 heads). You will notice, that for 3 heads calculation, I choose the term with p^3. (i.e. p raised to power of 3). If I wanted the probability of 2 heads, I would compute based on the term with p^2.

P(3 heads) = 4* p^3 * q^1 = 4 * (.5^3) * (.5^1) = .2500 = 25.00%

The math confirms the the area of the bar for 3 heads. (Note: We need to get rid of the idea that Ch 3 Figure 7 is somehow right because some expert said so. The PDF was not given to me by the Prophets. I had to calculate each term & then I drew the figure. So the 25% we just computed merely confirms I did it right when I built the graph shown. When something is true, YOU should be able to confirm its truth by study, by thinking and by applying yourself. That is what this statistics course is about.

**Homework: Ch 3, Problem 1:** Use the method above to compute the probability for each of the 5 possible outcomes of a 4 coin throw. (We just did the 2nd bar for you. Do the math for the other 4 bars). E-mail your answers to your teacher.

**Homework: Ch 3 Problem 2:**

** Cats and Kittens:**A vetranarian researcher has been keeping records for years on the color of kittens born in his office. He wants to know if color of the kitten matches a random variable, or if some other factor is at work in determining the color. His goal is to compare the probability to the office records.

A mother cat has 4 kittens. If the probability of a kitten being born black is 25%, what is the probability she will have exactly 3 black kittens?

Ch 3 Figure 8

- First, check the problem to verify that a Binomial Analysis is appropriate. The Binomial criteria is provided at the beginning of this chapter. Do you think we can use the Binomial Equation? Do all four criteria seem to be met?
- Create the Pascal's triangle with a sufficient number of terms for this problem.
- Create the Binomial Equation (all 5 terms) that describes the problem.
- Choose the correct term. Fill in the probabilities and calculate the answer.
- What is the answer? What is the probability of getting exactly 3 black kittens?
- Does your answer agree with the PDF Ch 3 Figure 8?
- E-mail your steps and the answer to your instructor.

- For any problem in general, if the probability of success is 1%, what is the probility of failure?
- Write the PDF equation for 6 bolts. Each bolt has a 1% chance of failure. Do not bother solving probabilities, just write the equation.
- E-mail your answers to your instructor.

**Optional: Where are the Realistic Examples? ** you might ask. Ch3 Problem 5 is pretty realistic. Precisely that kind of analysis is used for important aircraft structures where failure could result in loss of aircraft. The analysis is also performed for multiple computers that check each other during flight- but the issue here is probability of a computer breaking during the flight. Last, we will make great use of the binomial distribution during our development and explanation of the Weibull Probability Distribution. That is, some of the procedural methods in performing a Weibull Analysis are based on Binomial Analysis.

Contact the author
paul-watson@sbcglobal.net
by e-mail.

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Paul F. Watson

Beginning of St. Paul's Statistics Introduction

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