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Question 1: - ( a) het, X : no of answers students do correct. : probability of a correct answer = 1 / Since , all to questions would be independent of each other X~ B( , P ) where , n = lo p =1 / = 0.2 -Since , the firmif of Binomial random variable is , P ( X = x ) = " (x / * qn -x P ( X = x ) = 10 cx ( 0.2 ) " ( 0 . 8 ] to - x where, to ly = tol x 1 ( 10 - x ) 1 Since , Ite student need atleast 7 answers to pass the best to the probability that the student pass the test is, P (X 7 7 ) = P ( X = 7) + P ( x = 8) + P (x= q) + P ( x = 10 ) P ( x 7, 7 ) = 10 ( 7 ( 0 . 2 ) ? ( 0 . 8 ) 3 + 10 co ( 0 . 2 ) 8 6 0 . 8 ) 2 + 10 (9 ( 0 . 2 ) 160. 8 ) 1 + 10 ( 10 ( 0. 2 ) 10 ( 0 . 8 ) " P( x 7 /7 ) 0 . 00 086( b) let, X: no of requests for truck driver : mean no of requests = 2 perday X~ P ( ) where d = 2 The ping of Poisson distribution is, p ( x = x ) = ed d x where 1 = 2 P ( X = x) = e-2 (2)* Thes, the probability that ona given day these are atmust two request for truck drivers, P ( X = 2 ) = P ( x = ) + P ( = 1) + p (x = 2 ) P ( x = 2) = e ( 2 )" + e - ( 2 )' + e- 2 ( 2 ) o ! 1 1 21 P ( x = 2 ) = 5 e - 2 P ( X = 2 ) = 0. 67 66 8Binomial and Poisson Distribution 1. a. Some students do no revision, so they always must guess the answers in multiple-choice examinations. There are 10 questions, each with 5 possible answers. The students need at least 7 correct answers to pass the test. Find the probability that the students pass. [4 marks] b. On average a hardware store receives requests to hire out two truck drivers a day. Find the probability that on a given day there are at most two requests for truck drivers. [4 marks]