What is a Binomial distriubtion? Provide two examples for a binomial distribution.

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A binomial distribution is a probability distribution for cases that follow certain descriptions:

- There are 2 possible outcomes (called the "success" and "failure")

- The outcomes are independent, where the outcome of 1 trial is not dependent on what happens in the other trials

- The trials can be repeated

- The probability of success is the same every time

It can be calculated using P = nCr * p r (1-p)n-r

Where:

  n = Number of events
  r = Number of successful events
  p = Probability of success of one trial 
  1-p = Probability of failure (sometimes called q)
   nCr = ( n! / (n-r)! ) / r!

A classic example of the binomial distribution is the probability of flipping a certain number of heads or tails when flipping a coin, because there are 2 outcomes, one outcome does not affect the next, and it can be repeated as many times as you want without the probablity of "success" (ex: getting heads on one flip) ever changing from 1/2. 

Example: You toss a coin 13 times. What is the probability of getting exactly 6 heads?

(number of trials n = 13; number of successes r = 6 since we call getting a head as success; probability of success on any single trial p = 0.5)

Plug these numbers in for P = nCr * p r (1-p)n-r

Calculate nCr:

= ( n! / (n-r)! ) / r!
= ( 13! / (13-6)! ) / 6!
= ( 13! / 7! ) / 6!
 = ( 1235520 ) / 720
= 1716

P = 1716 * .5 6 (.5)7 

= .209472.. = 20.9%

Another common example of binomial distribution is with throwing a dice. Example: if a dice is thrown 5 times, what is the probability of there being 1 throw resulting in a 4? (We call getting a 4 a success)

Here,   5 = Number of events
  1 = Number of successful events
  1/6 = Probability of success of one trial 
  5/6 = Probability of failure (sometimes called q)
   nCr = ( 5! / (5-1)! ) / 1!

Hope that helps!

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