Binomial Distribution: Understanding the Basics, Examples, and Calculations, Slides of Statistics

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Binomial Distribution

The binomial distribution is a

discrete distribution.

Binomial Experiment

• A binomial experiment has the following properties:
  • experiment consists of n identical and independent trials
  • each trial results in one of two outcomes: success or failure
    • P(success) = p
    • P(failure) = q = 1 - p for all trials
  • The random variable of interest, X, is the number of successes in the n trials.
  • X has a binomial distribution with parameters n and p

Examples when assumptions do not hold

• Basketball player shoots ten free throws
  • Feedback affects independence and constant p
• Barrel contains 3 red apples and 4 green apples;

select 4 apples without replacement; X = # of red apples.

• Without replacement implies dependence

What is P(x) for binomial?

x n x

p q

x n x

n

P x

Example

• Random Guessing; n = 100 questions.
  • Probability of correct guess; p = 1/
  • Probability of wrong guess; q = 3/
  • Expected Value =
    • On average, you will get 25 right.
• Standard Deviation =

1 np 100 25 4

     (^)     

1 3 npq np 1 p 100 4. 4 4

     ^ ^        

Example

• Cancer Treatment; n = 20 patients
• Probability of successful treatments; p = 0.
• Probability of no success; q =?
• Calculate the mean and standard deviation.