 Hi and welcome to the session. Let us discuss the following question. Question says, it is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective? Let us now start with the solution. Now let us assume that x denotes the number of defective articles in 12 articles drawn randomly. So we can write, let x denotes the number of defective articles in 12 articles selected randomly. Now event of selecting random articles is independent and given number of trials is 12 or we can say number of trials is definite. So the trials are Bernoulli trials. Now we can write, since the event of selecting random articles is independent and the number of trials is definite, that is 12, the trials are Bernoulli trials. Now clearly we can see x has binomial distribution with n is equal to 12 and p is equal to 10 upon 100 which is further equal to 1 upon 10. We know 10% of certain articles manufactured are defective. So probability of getting a defective item or defective article is equal to 10 upon 100. We know 10% is equal to 10 upon 100. Now simplifying further we get p is equal to 1 upon 10. Now we will find out q, q is equal to 1 minus p, q represents the probability of getting good article. Now it is equal to 1 minus 1 upon 10 which is further equal to 9 upon 10. Now we know for binomial distribution v and p probability of x successes is equal to ncx multiplied by q raise to the power n minus x multiplied by p raise to the power x where x is equal to 0, 1, 2, till n. Now in the given question success is that item is defective. So we have to find the probability of 9 successes. We know total number of trials made is equal to 12 so we will substitute 12 for n here and value of x is 9, value of q is 9 upon 10 and here we will substitute 12 for n and 9 for x so here we can write 12 minus 9. Now value of p is equal to 1 upon 10 so we will substitute 1 upon 10 for p and here we will write 9, we know value of x is 9. Now simplifying right hand side we get 220 multiplied by 9 upon 10 cube multiplied by 1 upon 10 raise to the power 9. Now we can further write it as 220 multiplied by 9 cube upon 1000 multiplied by 1 upon 10 raise to the power 9. Now this 0 and this 0 will get cancelled and we get 22 multiplied by 9 cube upon 10 square we can write 100 as 10 square multiplied by 1 upon 10 raise to the power 9. Now using the laws of exponents we get 22 multiplied by 9 cube upon 10 raise to the power 11, bases are same so powers will get added. So we get probability that in a random sample of 12 articles 9 are defective is equal to 22 multiplied by 9 cube upon 10 raise to the power 11. This is our required answer this completes the session hope you understood the solution take care and have a nice day.