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)