Probability Distributions: Binomial Distribution

Question 1

How to know it it’s a binomial experiment?

Answer 1

1. Is there a fix number of trials? Yes
2. Are there only 2 possible outcomes? Yes
3. Are the outcomes independent of each other? Yes
4. Does the probability of success remain the same for each trial? Yes

Question 2

What is Binomial Distribution?

Answer 2

• A Binomial Distribution describes the probability of an event that only has 2 possible outcomes.
  
  • Ex. Heads or tails
  • Success or failure
• It can be used to describe the probability of a series of independent events that only have 2 possible outcomes occurring.
  
  • Ex. Flipping a coin 10 times and having it land with 5 on heads exactly 5 times. Or flipping a coin 10 times and having at least 5 land on heads.
• The probability of an event with n trails and f failures follows a binomial distribution
  
  • Visual graphed Binomial Distribution attached

Question 3

When Would You Use Binomial Distribution?

Answer 3

• Requirements and Conditions for Binomial Distribution
  
  • Must be a fixed number of trials
  • Discrete Data
    
    • Continuous data are NOT Binomial
  • Probability of success should be the same on every trial
  • Probability of success is constant
  • Two state. Two possible outcomes.
    
    • True or False, Hot or Cold, Success or Failure, Defective or Not Defective
  • Independent trails - trails are statistically independent
    
    • Ex. The flip of one coin means nothing to the results of the next coin flip
  • Use Binomial Distribution when you are sampling with replacement

Question 4

You Cannot Use Binomial Distribution On:

Answer 4

• When the probability of success is not constant for an event
  
  • Ex. The probability of it snowing or not showing in NYC would not fit the criteria for a Binomial Distribution because the probability of success is not constant. The chance of snow on winter days is higher than summer days
• When you sample without replacement
  
  • In that cause, use Hypergeometric

Question 5

Cumulative vs Non-Cumulative

Answer 5

• There are 2 ways I’ve seen Binomial Distribution problems represented in Six Sigma Exams:
  
  • Non-Cumulative questions
  • Cumulative questions (with or without a chart)
• The question can either be about the actual equations and translating a word problem into an actual solution
  
  • OR
• The questions can be about the necessary scenarios when you would apply binomial probability
  
  • In other words, what MUST be true to use it, or if an event is binomial, what can we infer about it’s properties.

Question 6

Non Cumulative Probability

Answer 6

• This is just a fancy statistics term of single events taken on their own
  
  • Example:
    
    • A single flip of a coin

Question 7

Cumulative Probability

Answer 7

• Fancy statistics term for multiple events.
  
  • I like to remember that Cumulative events literally accumulate multiple probabilities.
• One other thing to be away of is that you can be asked to either manually calculate a binomial cumulative probability question or could be used to leverage a chart.

Question 8

Notes About Sampling with Replacement

Answer 8

• Binomial is for N trials and F failures and describes the probability of k successes in n draws with replacement from a finite population of size N containing exactly K successes.
• Remember, binomial is for independent events where the likelihood of each event occurring is the same as the others. If you sample without replacing, you’re affecting the likelihood of independence

Question 9

Non-Cumulative Binomial Probability

Calculation Example 1 “Contains Exactly”

Answer 9

• If a coin is tossed 10 times, what is the probability of obtaining exactly four heads?
  
  • With coin tosses there are only 2 possible outcomes; heads or tails. So you should be on alter for using Binomial.
    
    • n = the number of trails or the sample size
    • x = the specified number of “successes” in the entire sample
    • p = the probability that the specified result will occur on a single trial or sample (a “success”)
    • Success Heads
      
      • n = 10
      • x = 4
      • p = 0.5

Question 10

Non-Cumulative Binomial Probability

Calculation Example 2 “Contains Exactly”

Answer 10

• A manufacturing process creates 3.4% defective parts. A sample of 10 parts from the production process is selected. What is the probability that the sample contains exactly 3 defective parts?
  
  • As soon as you see the word defective, you should be alert to using the Binomial equation. Since defect in this sense means that a part is in a binary state, either function of defective, it meets our criteria.
    
    • n = the number of trails, or the sample size
    • x = the specified number of “successes” in the entire sample
    • p = the probability that the specified result will occur on a single trail or sample (a “success”)
    • n = 10
    • x = 3
    • p = 0.034

Question 11

Non-Cumulative Binomial Probability

Calculation Example 3 “Contains Exactly”

Answer 11

• Forty-five percent of all registered voters in a national election are female. A random sample of 8 voters is selected. The probability that the same contains 2 males is:
  
  • Since in this question the assumption is that there are (2) biological genders measures, male or female, you should be on alert to use the binomial distribution
    
    • n = number of trails, or the sample size
    • x = the specified number of “successes” in the entire sample
    • p = the probability that the specified result will occur on a single trial or sample (a “success”)
    • n = 8
    • x = 2
    • p = 0.55 (because 45% are female, therefore the remaining 55% are male)

Question 12

Cumulative Binomial Probability Calculation Example

Example 1

“At least” or “Fewer Than” or “At Most”

Answer 12

• 79% of the students of a large class passed the final exam. A random sample of 4 students are selected to be analyzed by the school. What is the probability that the sample contains fewer than 2 students that passed the exam?
  
  • Since we are examining data that only has a binary state, pass or fail, you should be on alert to using the binomial equation.
  • Also, since you’re asked for ‘x or fewer,’ you have to calculate the probability of ALL POSSIBLE events.
  • In this example, we must calculate the odds of EXACTLY one pass PLUS the odds of EXACTLY no pass in the sample.
    
    • n = the number of trails, or the sample size
    • x = the specified number of “successes” in the entire sample
    • p = the probability that the specified result will occur on a single trail or sample ( a “success”)

Question 13

Cumulative Binomial Probability Calculation Example

Extension of Example 1

“More than”

Answer 13

An example of this kind of question is “Test the probability of the number of dropped calls will exceed a certain number.”

Here you’d do the opposite of the “Fewer” example above and count the probabilities above.

So, if we re-wrote the question above to be ‘What’s the probability of sampling 10 students and 8 or more passed’ it would be the probability of EXACTLY 8 passing, PLUS EXACTLY 9 passing, PLUS EXACTLY 10 passing.

Question 14

Binomial Chart Question Example: (Without Calculations)

Answer 14

• If I were using a Binomial Distribution Table, which values of X would I add together to get the probability of at most N defective?
  
  • You would add the probabilities of N, n-1, n-2… all the way to zero.
  • Ex. “which values of X would I add together to get the probability of at most 4 defective?”
    
    • Add the probabilities of 4 + 3 + 2 +1 + 0.

Question 15

Binomial Chart Question Example: (“More Than”)

Answer 15

Question 16

Example of Using Binomial Probability in a Six Sigma Project

Answer 16

• Typically in a Six Sigma project you are using DMAIC methodology. Let’s say during the Define Phase you charter a project to see if you could improve a process that had a binomial outcome. You might even use the Binomial Distribution to articulate the business case for why you should do the project in the first place - for instance, event A should be binomial, but it’s clearly not.
• You might use the Binomial distribution is see if sample of the process outcome were in face following a binomial distribution during the Analysis phase
• If you are running a pilot implementation during the Improve Phase, you might make the assumption that some events that are inputs to your process in face will follow a binomial distribution.
• In the Control Phase, you might monitor that specific process is generating Binomially distributed results if it should.

Question 17

Binomial Distribution Formula and Example

Answer 17

Question 18

Main Characteristics of a Binomial Distribution

Answer 18

• The experiment involves n identical trails
• Each trial has only two possible outcomes denoted as success or failure
• The trails are independent of each other
• Denote p as the probability of success, which remains the same between trails, and q = (1 - p) as the probability of getting a failure on any trail