Raffles Institution H2 Math S2B Binomial Distribution Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _______________________________ Tutorial S2B: Binomial Distribution Page 1 of 3 Tutorial S2B: Binomial Distribution Section A (Discussion Questions) 1 State, with a reason, whether a binomial distribution could be used in each of the following problems. If binomial distribution is an acceptable model, define the random variable clearly and state its parameters. (a) A gambler has a biased coin for which the probability of obtaining a head in any toss is 0.56. Find the probability of getting 6 heads if he tosses the coin 8 times. (b) A fair coin is spun until a head occurs. Find the probability that 8 spins are necessary, including the one on which the head occurs. (c) A jar contains 49 balls numbered 1 to 49. Six of the balls are drawn at random without replacement. Find the probability that four out of the six balls drawn have an even score. 2 Given ~B 1 0 ,0 . 2X , find (i) ; (ii) ; (iii) ; (iv) P(4 8)X ; (v) P(2 6)X ; (vi) E X ; (vii) Var X . [ (i) 0.107 (ii) 0.678 (iii) 0.0328 (iv) 0.121 (v) 0.618 (vi) 2 (vii) 1.6] 3 A die is biased such that the probability of getting a six is 0.18. If the die is thrown ten times, find (i) the probability that the number of sixes that turn up is odd, (ii) the expectation and variance of the number of sixes that turn up. [(i) 0.494, (ii) 1.8, 1.476] 4 Two fair dice are thrown. Find the probability that the total score is two-figured. If five people were each to throw two dice, find the probability that exactly three of them will get a two-figured total score. [ 1 6 , 0.00322] 5 The probability that a patient suffering from a particular disease will be cured following a new treatment is 0.92. (i) If the treatment is given to 10 patients, show that the probability that at least 9 out of the 10 patients will be cured is 0.812, correct to 3 significant figures. (ii) This treatment is being tested at 12 hospi tals where in each hospital, the treatment is given to 10 patients. Find the probabil ity that more than 10 of the 12 hospitals will have a success rate of at least ninety percent. [(ii) 0.310] P( 0)X  P( 2)X  P( 5)X 

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Tutorial S2B: Binomial Distribution Page 2 of 3 6 According to the school rules, a student who arrives at school after 0730 hours is considered late. (i) During the school term, a boy cycles to sc hool on 5 days each week. On any given day, the probability that he arrives after 0730 hours is 0.1. For a period of 4 weeks, calculate the probability that he is late on at least one day but not more than 3 days. When the boy has cycled to school for the nth time, the probability that he arrived at school late at least once is greater than 0.99. Calculate the least value of n. (ii) A girl travels to school by bus on 5 days a week. Over a long period of time, the variance of the number of days per week on which she is late is 0.8. Given that p denotes the probability that she arrives late and that 0.5p , evaluate p. [(i) 0.745, 44 (ii) 0.2] 7 A female doctor wishes to test a new treatment for a certain disease. She asks patients who come to her clinic suffering from this dis ease if they are willing to take part in a trial. Based on past experiences, she finds that the probability of such a patient willing to take part is 0.3. (i) Show that the probability that at least 2 pati ents are willing to take part in this trial when she has asked 8 patients is 0.745, co rrect to 3 significa nt figures. What assumption do you have to make to justify the use of the distribution in your working? (ii) The doctor decides to ask n patients. Find the least number of patients she has to ask so that she will have a probability of more than 0.8 that at least 2 patients are willing to take part in this trial. (iii) Find the most likely number of patients w ho are willing to take part in this trial when she asks 9 patients. (iv) State the expected number of patients who are willing to take part in this trial when she asks 9 patients. [(ii) 9 (iii) 2 and 3 (iv) 2.7] 8 In a promotional campaign organized by the management of the petrol filling station, a customer who refills his car with at least $50 worth of petrol is entitled to draw 5 tickets from a lucky draw box at random. 1 6 of the tickets in the lucky draw box are winning tickets while the rest are non-winning tickets . The 5 tickets drawn by a customer are put back into the box before the draw by the next customer. State an assumption needed for a binomial distribution to be a valid model for the number of winning tickets drawn in a draw of 5 tickets. Using a binomial distribution as the va lid model, find the probability that among 3 randomly chosen customers who participated exactly once in a draw, only one of them obtained at least one winning ticket in the draw of 5 tickets. [ 0.290]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Tutorial S2B: Binomial Distribution Page 3 of 3 9 National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of 25. It is known that 100 p% of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is 1. (i) Explain why the context above may not be well-modelled by a binomial distribution. [1] Assume now that the context above is well-modelled by a binomial distribution. (ii) State the value of p. [1] (iii) Find the probability that a box chosen at random has less than 2 rotten tomatoes. [2] (iv) A customer chose a box and inspected the contents individually. Find the probability that the twenty-first tomato is the fourth rotten tomato and no rotten tomatoes are found subsequently. [3] Boxes that contains at least 24 tomatoes that are not rotten are deemed satisfactory. (v) A customer first picks 3 boxes of to matoes, of which at least 2 boxes are satisfactory. The customer then decides to buy another 5 boxes. Find the probability that exactly 6 of the 8 boxes are satisfactory. [4] [(ii) 0.04 (iii) 0.736 (iv) 0.00124 (v) 0.335] 10 In a computer game. a bug moves from left to right through a network of connected paths. The bug starts at S and, at each junction, randomly takes the left fork with probability p or the right fork with probability q , where 1.qp The forks taken at each junction are independent. The bug finishes its journe y at one of the 9 endpoints labelled A  I (see diagram). (i) Show that the probability that th e bug finishes its journey at D is 5356 .pq [2] (ii) Given that the probability that the bug fi nishes its journey at D is greater than the probability that the bug finishes its journey at any one of the other endpoints, find exactly the possible range of values of p . [4] In another version of the game, the probabil ity that, at each junction, the bug takes the left fork is 0.9 p , the probability that the bug takes the right fork is 0.9q and the probability that the bug is swallowed up by a ‘black hole’ is 0.1. (iii) Find the probability that, in this version of the game, the bug reaches one of the endpoints A I, without being swallowed up by a black hole. [1] [(ii) 52 93 p (iii) 0.430 ]