I am facing problems while comparing the results of solving a problem individually using both the concept of Binomial Distribution of Probabilities and the Classical Definition of Probability. Let me formulate the problem first: "The probability that a pen manufactured by a company will be defective is 1/10.If 12 such pens are manufactured find the probability that exactly 2 pens are defective." Solution: If 'X' be a random variable representing the number of defective pens , then, 'X' is a bivariate with parameters n=12 and p=1/10 where q=1-p=9/10. For exactly 2 defective pens out of 12 manufactured pens we should have the probability of such an event as, P(X=2)=12C2.p2.q12-2=0.2301 Now,if we try to solve the problem in a different way using the classical definition of probability, we have n(S)=2n=212=4096 and, n(E)=nCr=12C2=66, where 'E' is the event of obtaining 'r' successes in 'n' independent trials i.e, 'r' defective pens among 'n' manufactured pens in case of this particular problem. So P(E)=n(E)/n(S)=66/4096=0.016. Thus, from the classical definition of probability we obtain the probability of 'r' successes in 'n' independent trials of an experiment as something different from what we obtained using the theory of binomial distribution of probability. Can anybody explain where did I go wrong?