Binomial Distribution for Successive Events

        As experiment may be repeated n times and each of these n trials are independent of one another. If each trial only gives one of the two possible outcomes, say 'success' (when required event take place) and failure (when required event does not take place) then it is called a Binomial Experiment. Let probability of success in any trial be p and that of failure be q, then
p + q = 1        Then (p + q)ⁿ = C₀ Pⁿ + C₁P^n-1q +...... C_rp^n-rq^r +...+ C_nqⁿThen the probability of exactly k successes in n trials is given by        P_k = ⁿC_kq^n-kp^k        Some important facts related with binomial distribution        (i)     The probability of getting at least k successes is                P(x > k) = Σⁿ_x=k  ⁿC_x p^x q^n-x         (ii)    Σⁿ_x=k  ⁿC_x q^n-x p^x = (q + p)ⁿ = 1Illustration:        In a rainy season, there is 60% chance that it will rain on a particular day. What is the probability that there will exactly 4 rainy days in a week?Solution:        Let probability of raining on a particular day P(success) = 3/5        And probability that no rain on a particular day q(failure) = 2/5        Probability or exactly 4 rainy day i.e. 4 success out of 7 trials

                      = ⁷C₄ (p)⁴(q)³ =  ( 7!/4!.3! ) * ( 3/5 )⁴ * ( 2/5 )³