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Binomial Distribution Tutorial

  


Definition:

          The Binomial Distribution is one of the discrete probability distribution. It is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled Success and Failure. The Binomial Distribution is used to obtain the probability of observing r successes in n trials, with the probability of success on a single trial denoted by p.


Formula:

P(X = r) = nCr p r (1-p)n-r
where,
             n = Number of events
             r = Number of successful events.
             p = Probability of success on a single trial.
             nCr = ( n! / (n-r)! ) / r!
             1-p = Probability of failure.


Example: Toss a coin for 12 times. What is the probability of getting exactly 7 heads.


  Step 1: Here,
                Number of trials n = 12
                Number of success r = 7 since we define getting a head as success
                Probability of success on any single trial p = 0.5

  Step 2: To Calculate nCr formula is used.
           nCr = ( n! / (n-r)! ) / r!
                = ( 12! / (12-7)! ) / 7!
                = ( 12! / 5! ) / 7!
                = ( 479001600 / 120 ) / 5040
                = ( 3991680 / 5040 )
                = 792

  Step 3: Find pr.
            pr = 0.57
               = 0.0078125

  Step 4: To Find (1-p)n-r Calculate 1-p and n-r.
            1-p = 1-0.5 = 0.5
            n-r = 12-7 = 5

  Step 5: Find (1-p)n-r.
            = 0.55 = 0.03125

  Step 6: Solve P(X = r) = nCr p r (1-p)n-r
            = 792 * 0.0078125 * 0.03125
            = 0.193359375


The probability of getting exactly 7 heads is 0.19



Related Calculator:
>> Binomial Distribution Calculator

This tutorial will help you to calculate the Binomial Distribution problems.