Binomial Distribution Qns and Solns

Chapter 3: Binomial Distribution 1. JPJC JC2 Prelim 8865/2019/Q6 Mandarin oranges are supplied to the supermarkets in boxes of 12. It is found that on average, 7% of the mandarin oranges are bad when the boxes are opened. (i) For a randomly chosen box of mandarin oranges, find the probability that (a) all the mandarin oranges are good, [1] (b) at least 25% of the mandarin oranges are bad. [2] (ii) A customer bought 20 boxes of mandarin oranges. Find the probability that fewer than 4 boxes have at least 25% bad mandarin oranges. [2] Answer: (ia) 0.419, (ib) 0.0468, (ii) 0.987 1. JPJC JC2 Prelim 8865/2019/Q6 (Solutions) (ia) Let X be the random variable denoting the number of bad mandarin oranges out of 12 B(12,0.07)X P( 0) 0.418596 0.419 (3sf)X = =  Alternatively, Let X’ be the random variable denoting the number of good mandarin oranges out of 12 ' B(12,0.93)X P( ' 12) 0.418596 0.419 (3sf)X = =  (ib) P( 0.25 12) P( 3) 1 P( 2) 1 0.95320 0.046797 0.0468 (3sf) XX X   =  = −  =− = (ii) Let Y be the random variable denoting the number of cartons out of 20 with at least 25% bad mandarin oranges. B(20,0.0467968)Y P( 4) P( 3) 0.98728 0.987 (3sf)YY =  =  2. MI PU2 Prelim 8865/2019/Q10 Millennia Tour operates a half-day city tour on an open double-decked bus. A bus has 22 seats and Millennia Tour will accept a maximum of 22 passenger bookings per bus. Past records show that on average 30% of the passengers do not turn up for the tour aft er booking. Tickets sold are non - refundable and non-exchangeable. The tour is always fully booked on weekends. Find the probability that, in a bus on a randomly chosen Saturday, (i) there are exactly 7 passengers who do not turn up for the tour after booking, [1] (ii) there are at least 17 passengers who turn up for the tour after booking. [2]

(iii) 8 open double -decked buses are fully booked on a randomly chosen Saturday. Find the probability that at least 3 of these buses have at least 17 passengers who turn up for the tour after booking. [3] During holiday peak seasons, Millennia Tour decides to maximise profit by accepting n bookings per bus on weekends, where 22n . (iv) The probability that sufficient seats are available for all passengers who turn up for the tour on a randomly chosen Saturday during a holiday peak season is more than 0.9. Find the largest value of n. [3] Answer: (i) 0.177 (ii) 0.313 (iii) 0.482 (iv) 27 2. MI PU2 Prelim 8865/2019/Q10 (Solutions) (i) Let W be the random variable denoting the number of passengers who do not turn up for the tour after booking, out of 22. Then B(22,0.3)W . ( )P 7 0.17707 0.177W = = = (ii) Method 1 Let Q be the random variable denoting the number of passengers who turn up out of 22. B(22,0.7)Q . ( ) ( )P 17 1 P 16 0.31341 0.313QQ = −  = = Method 2 ( )P 5 0.31341 0.313W  = = (iii) Let X be the random variable denoting the number of buses with at least 17 passengers who turn up for the tour, out of 8. B(8,0.31341)X ( ) ( )P 3 1 P 2 0.48216 0.482XX = −  = = (iv) Let Y be the random variable denoting the number of passengers who turn up, out of n passengers (bookings). Then B( ,0.7)Yn . ( )P 22 0.9Y  Using GC, Hence the maximum bookings should be 27. n ( )P 22Y  26 0.974 27 0.9409 28 0.8872

3. HCI JC2 Prelim 8865/2019/Q10 At the coffee and tea stall in the canteen of Hwa Chong Institution, on average, 70% of the students will order ice lemon tea. A random sample of n students are interviewed on their preference. (i) State in context, two assumptions needed for the number of students who will order ice lemon tea to be well modelled by a binomial distribution. [2] Assume now that the number of students who will order ice lemon tea follows a binomial distribution. It is given that 15n= for the rest of the question. (ii) Find the probability that exactly 10 students will order ice lemon tea. [1] (iii) Find the probability that more than 9 students will order ice lemon tea. [2] (iv) If the probability of at least r students ordering ice lemon tea is not less than 0.8, find the greatest value of r. [3] Answer: (ii) 0.206 (iii) 0.722 (iv) 9 3. HCI JC2 Prelim 8865/2019/Q10 (Solutions) (i) Assumptions: 1. The probability of students ordering ice lemon tea is constant at 0.7 for every student. 2. The order of a student is independent of the order of another student. (ii) Let X be the random variable denoting the number of students ordering ice lemon tea (out of 15 students). ( )B 15,0.7X ( )P 10 0.20613 0.206X = =  (3 s.f.) (iii) ( ) ( ) P9 1 P 9 0.721621 0.722 X X  = −  =  (iv) ( )P 0.8 1 P( 1) 0.8 P( 1) 0.2 Xr Xr Xr  −  −   −  From G.C., P( 9 1) 0.1311 0.2 P( 10 1) 0.2784 0.2 X X  − =   − =  1 9 1 9rr − = −  = Hence the greatest value of r is 9.

4. EJC JC2 Prelim 8865/2019/Q7 In a large number of apples, 2% of the apples are rotten. Uncle Ee buys cartons of apples from the fruit wholesaler every day. He randomly selects 6 apples from a carton and accepts the carton if all 6 are not rotten. (i) Explain the significance of the phrase ‘large number’ in the first sentence of the question. [1] (ii) Find the probability that Uncle Ee accepts a carton of apples. [1] (iii) Uncle Ee first picks 3 cartons of apples, of which at least 2 cartons are accepted. He then decides to buy another 5 cartons. Find the probability that exactly 6 of the 8 cartons are accepted. [4] Answer: (ii) 0.886 (iii) 0.163 4. EJC JC2 Prelim 8865/2019/Q7 (Solutions) (i) This is to assume that the conditional probability of the event that each apple (among the ones in each carton) is rotten is approximately the same and hence the trials (selection of apples) may be considered to be independent. (ii) ( ) ( ) ( ) Let be the number of apples, out of 6, that are rotten. B 6,0.02 P carton is acc random variable d epted P 0 enoting 0.885 th 84 0.886 (3sf e ) X X X== == Alternatively, Let Y be the random variable denoting the number of apples, out of 6, that are not rotten ~ B(6,0.98)Y ( ) ( )P carton is accepted P 6 0.88584 0.886 (3sf) Y== == (iii) ( ) Let be the number of cartons, out of fi rst 3, that are accepted. B 3,0 random variable denoting the random variable .88584 Let be the number of cartons, out of ne xt 5, denoting that are athe ccepted Q Q R ( ) . B 5,0.88584R

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P exactly 6 out of 8 accepted|at least 2 out of first 3 cartons accepted P 2, 4 P 3, 3 P2 P 2 P 4 P 3 P 3 1 P 1 0.163 3sf Q R Q R Q Q R Q R Q = = + = ==  = = + = == − = 5. NJC JC2 Prelim 8865/2019/Q7 A survey was conducted with a large number of young employees who were asked to select one SkillsFuture programme that they would like to enroll in to further develop their skills. The survey showed that 53% would like to enroll in infocomm technology programme, 38% would like to enroll in financial management programme and 9% would like to enroll in human resource programme. Fifteen young employees who took part in the survey were randomly selected. Find the probability that (i) exactly twelve young employees said that they would like to enroll in either infocomm technology programme or financial programme, [2] (ii) no less than five young employees said that they would like to enroll in human resource programme. [2] Suppose now that there are n young employees who took part in the survey were randomly selected, write down an inequality in terms of n such that the probability of thirty or thirty -one young employees said that they would not like to enroll in fina ncial management programme is at least 0.22. Hence, find the least value of n. [3