Binomial Distribution

Photo by Nico Buculei

Photo by Nico Buculei

A Binomial Distribution only has 2 possible outcomes, including replacement. Ex. Heads or tails.

Binomial is for N trials and F failures.

Describes the probability of k successes in n draws with replacement from a finite population of size N containing exactly K successes.

This is in contrast to the hypergeometric distribution, which describes the probability of k successes in n draws when sampling without replacement.

Requirements and Conditions for a Binomial Distribution

  • Must be a fixed number of trials.
  • 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 trials – trials are statistically independent.
  • Use Binomial Distribution when you are sampling with replacement.

Examples: http://www.six-sigma-material.com/Binomial-Distribution.html

You Cannot Use Binomial Distribution On:

  • The probability of it snowing or not snowing 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.

Binomial Probability Distribution

Binomial Probability Distribution

These come in 2 variations: Cumulative & Non-Cumulative.

Cumulative

  • Order of when they happen does matter.

Non Cumulative

  • Order of when they happen does not matter.

http://www.six-sigma-material.com/Binomial-Distribution.html

Non Cumulative Binomial Probability

Non Cumulative Calculation Example: Coin toss problem

If a coin is tossed 10 times, what is the probability of obtaining exactly four heads? (If an honest coin is tossed 10 times, what is the probability of 4 consecutive heads?)

binomial successes

binomial successes

 

Non Cumulative Calculation Example: Probability of exactly defective parts.

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?

binomial successes 2

binomial successes 2

Non Cumulative Calculation Example: Probability sample contains 2 males.

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 sample contains 2 males is:

binomial successes 3

binomial successes 3

 

79% percent 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?

binomial successes 4

binomial successes 4

 

 

Non-Cumulative Chart Example: 

You can see that the same percentage was found on the chart. (In this case, x is the probability of the event happening instead of p.)

 

 

Non-Cumulative Chart Example

 

More Non-Cumulative Questions:

  • “At Most”
    • If I were using a non-cumulative 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.

 

 Additional Binomial Distribution Questions:

  • Test the probability of the number of dropped calls will exceed a certain number.

 

Cumulative Binomial Probability

 

Good online calculator here:

Cumulative Calculation Example:

ddd

Cumulative Chart Example:

 

Additional Questions

The probability of an event with n trials and f failures follows a binomial distribution.

Additional Reference

2 comments… add one
  • Amy

    I think there is a typo:

    Cumulative

    Order of when they happen does not matter.

    Non Cumulative

    Order of when they happen does not matter.

    did you mean:

    Cumulative

    Order of when they happen does matter.

    Non Cumulative

    Order of when they happen does not matter.

    Thanks for building this resource!

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