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Negative Binomial Distribution Tutorial

  


Definition:

          The probability distribution of a Negative Binomial random variable is called a Negative Binomial Distribution. It is also known as the Pascal distribution or Polya distribution. Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a Negative Binomial Experiment.

Formula:

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


Example: Find the probability that a man flipping a coin gets the fourth head on the ninth flip.


  Step 1: Here,
          Number of trials n = 9 (because we flip the coin nine times).
          Number of successes r = 4 (since we define Heads as a success).
          Probability of success for any coin flip p = 0.5

  Step 2: Find n-1 and r-1.
         n-1 = 9-1 = 8
         r-1 = 4-1 = 3

  Step 3: To find n-1Cr-1 Calculate ((n-1)-(r-1))!
         (n-1)-(r-1) = 8-3 = 5
         ((n-1)-(r-1))! = 5! = 120

  Step 4: Find (n-1)!
          = 8! = 40320

  Step 5: Find (r-1)!
          = 3! = 6

  Step 6: Find (n-1)! / ((n-1)-(r-1))!
          = 40320/120 = 336

  Step 7: To Solve n-1Cr-1 formula is used.
          = 336/6 = 56

  Step 8: Find pr.
          = 0.54 = 0.0625

  Step 9: To Find (1-p)n-r Calculate 1-p and n-r.
         1-p = 1-0.5 = 0.5
         n-r = 9-4 = 5

  Step 10: Calculate (1-p)n-r.
          = 0.55 = 0.03125

  Step 11: Calculate Negative Binomial Distribution.
          = 56×0.0625×0.03125 = 0.109375


The probability that the coin will land on heads for the fourth time on the ninth coin flip is 0.1094.



Related Calculator:
>> Negative Binomial Distribution Calculator.

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