Binomial Distribution

In Statistics and Probability theory, the binomial distribution gives the discrete probability distribution, PP(n|N) for obtaining exactly n successes out of N trails (where the result of each Bernoulli trail is true with probability p and false with probability q= 1-p). In simple terms the binomial distribution describes the no. of times that a particular event will occur in sequence of observations. The event might be in binary form, i.e. it may or may not occur. It is used when the researcher wants to find out the occurrence of an even but not in the intensity or magnitude of the event. Say for ex: if the researcher wants to calculate the probability of raining. He will concentrate on rain but not on the damages that occur due to the rain.

The binomial distribution is therefore given by

where is a binomial coefficient. The above plot shows the distribution of successes out of trials with .

Example: Toss a coin for 10 times. What is the probability of getting exactly 8 heads.

  Step 1: Here,                 Number of trials n = 110                Number of success r = 8 (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!                = ( 10! / (10-8)! ) / 8!                = ( 10! / 2! ) / 8!                = ( 3628800 / 2 ) /                 = ( 1814400 / 40320 )                = 45

  Step 3: Find pr.            pr = 0.58

               = 0.00390625

  Step 4: To Find (1-p)n-r Calculate 1-p and n-r.            1-p = 1-0.5 = 0.5            n-r = 10-8 = 2

  Step 5: Find (1-p)n-r.            = 0.52 = 0.25

  Step 6: Solve P(X = r) = nCr p r (1-p)n-r

            = 45 × 0.00390625 × 0.25            = 0.0439453125

The probability of getting exactly 8 heads is 0.044