RI S2B Binomial Distributions Add Prac Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _____________________________________________________ Additional Practice Questions for Chapter S2B: Binomial Distribution Page 1 of 16 Additional Practice Questions for Ch apter S2B: Binomial Distribution (Solutions) 1 Published articles in medical journals indicate that, on average, 35 out of 100 patients having a lumbar puncture will suffer SSH (‘Severe Spinal Headache’). Twelve patients are given a lumbar puncture. (i) Using a binomial model, find the expected number of patients who will suffer SSH, and find also the standard deviation. [3] (ii) Find the probability that four or more of the twelve patients will suffer SSH. [2] Solution: (i) Let X be the number of patients, out of 12, who will suffer SSH. Assumptions: (1) Whether a patient suffers SSH is independent of another. (2) The probability of a patient suffering SSH remains constant at 0.35. Then ~ B(12,0.35)X Expected number of patients who will suffer SSH = 12(0.35) = 4.2 Standard deviation = 12(0.35)(1 0.35) 1.65 (3.s.f) (ii) P( 4) 1 P( 3)XX   =0.653 (3.s.f) 2 The random variable X ~ B (16, p), where p < 0.5. If the variance of X is 3.36, find the value of p. Find also the probability that X is less than its mean. [4] Solution: Variance = (16)(p)(1p) = 3.36, which gives p = 0.3 or 0.7 (rejected since p < 0.5) Hence, p = 0.3 Mean = 16(0.3) = 4.8 P( 4.8) P( 4)XX = 0.450 (3 s.f.)

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ _____________________________________________________ Additional Practice Questions for Chapter S2B: Binomial Distribution Page 2 of 16 3 TPJC Prelim 2007/02/Q5 A factory produces chocolate which are packed into boxes of 20 and delivered to shops for sale. A chocolate will not meet the minimum criteria for packing for sale if it weighs less than 20 grams. On average, 2% of the chocolate produced did not meet the minimum criteria. (i) Find the probability that a randomly chosen box contain at least 1 chocolate that does not meet the minimum criteria. [2] (ii) Find the probability that out of 4 randomly chosen boxes of chocolate, there are exactly 2 boxes with at least 1 chocolate that does not meet the minimum criteria. [2] Solution: (i) Let X be the number of chocolates, out of 20, not meeting the minimum criteria. Assumptions: (1) Whether a chocolate does not meet the minimum criteria is independent of another. (2) The probability of a chocolate not meeting the minimum criteria remains constant at 0.02. Then ~B ( 2 0 , 0 . 0 2 )X P( 1) 1 P( 0) 0.33239 0.332 (3 s.f.)XX    (ii) Let Y be the number of boxes of chocolates, out of 4, with at least 1 chocolate that doesn’t meet the minimum criteria. Then ~ B(4,0.33239)Y P( 2) 0.295Y  (3 s.f.) 4 CJC Prelim 2006/02/Q25 In a multi-national company with a large population, 13.5% of the staff owns a vehicle. (i) Find the probability that in a random samp le of 30 staff, exactly 5 of them will own a vehicle. [2] (ii) The probability that there is at least one st aff who owns a vehicle in a random sample of size n is greater than 0.95. Find the least value of n. [3] Solution: (i) Let X be the number of staff, out of 30, who owns a vehicle. Assumptions: (1) Whether a staff owns a vehicle is independent of another. (2) The probability of a staff owning a vehicle remains constant at 0.135. Then ~ B(30,0.135)X P( 5) 0.170X  (3 s.f.) (ii) Let Y be the number of staff, out of n, who owns a vehicle. ~B ( , 0 . 1 3 5 )Yn Then we have P( 1) 0.95 1P ( 0 ) 0 . 9 5 P( 0) 0.05 (1 0.135) 0.05 ln 0.05 20.7ln 0.865 21 n Y Y Y n n            Least value of n is 21. or using a GC, When n = 20, P(Y=0)  0.054995 > 0.05 When n = 21, P(Y=0)  0.047571 < 0.05

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ _____________________________________________________ Additional Practice Questions for Chapter S2B: Binomial Distribution Page 3 of 16 5 RVHS Prelim 2014/02/Q9 (part) Market research showed that 3 out of 10 hous eholds in a housing estate subscribe to fibre broadband internet services. (a) 20 households from a particular block of flats in the estate were surveyed. (i) Show that the probability of more th an 3 and less than 9 of the households surveyed subscribe to fibre broadband internet services is 0.780, correct to 3 decimal places. [2] (ii) Find the least value of k such that the probability that at most k of the households surveyed subscribe to fibre broadband internet services is at least 0.75. [2] (b) There are a total of 50 blocks of flats in the estate. 20 households from each block of flats were surveyed. Find the probability that there are exactly 39 blocks of flats with more than 3 and less than 9 households surveyed that subscribe to fibre broadband services. [2] Solution: (a)(i) Let X be the number of households that subscribe to broadband internet services out of 20 households. Assumptions: (1) Whether a household subscribe to broadband internet services is independent of another. (2) The probability of a household subscribing to broadband internet services remain constant at 0.3. ~B 2 0 ,0 . 3X P3 9 X P4 8 X   P8 P3XX       0.77958 5 d.p. 0.780 3 d.p. Shown   (ii) P0 . 7 5Xk Using the GC to set up a table of values, P6 0 . 6 0 8 0 . 7 5X    P 7 0.772 0.75X   The least value of kis 7. (b) Let Y be the number of blocks, with more than 3 and less than 9 households subscribing to broadband internet services, out of 50 blocks each with 20 households surveyed.  ~ B 50,0.780Y    P 39 0.135 3 s.f.Y

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ _____________________________________________________ Additional Practice Questions for Chapter S2B: Binomial Distribution Page 4 of 16 6 RI CT2/H1/2018/Q5 It is known that 36% of the customers of a certain supermarke t will bring their own environmental friendly bags. On a certain day, there are 3 cashiers and each cashier has 5 customers in queue. (i) Find the probability that among all the cust omers in queue, at least 4 of them brought their own environmental friendly bags. [2] (ii) If exactly 4 customers in queue brought thei r own environmental friendly bags, find the probability that each cashier will have at least 1 customer bringing his or her own environmental friendly bag. [4] Solution (i) Let X denote the number of customers (out of 15) who brought their own environmental friendly bags. X ~ B(15, 0.36) P(X  4) = 1  P(X  3) = 0.84694 = 0.847 (3 sf ) (ii) Let Y denote the number of customers (out of 5) who brought their own environmental friendly bags. Y ~ B(5, 0.36) P(Each cashier will have at least 1 customer bringing his or her own environmental friendly bag given that X = 4) = 3  P(Y1 = 1)  P(Y2 = 1)  P(Y3 = 2) P(X = 4) = 3  (0.30199)2  0.33974 0.16917 = 0.54945 = 0.549 (3 sf )

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ _____________________________________________________ Additional Practice Questions for Chapter S2B: Binomial Distribution Page 5 of 16 7 MJC Pre