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My Additional Mathematics Modules Form 5 (Version 2011) Topic: 20 DISTINCTION in by NgKL 2 20.1 BINOMIAL DISTRIBUTION Important Notes 3 Exercise 20.1.1 − Determine the Probability by Binomial Distributions P(X = r ) = n Cr p r q n - r 1. Madonna has bought 6 chicken eggs from a sundry 2. The probability that Jacob will win in a BSMM shop. The probability of getting rotten eggs from a competition is ⅜. If he participates in 5 events, sundry shop is 0.2. Find the probability that calculate the probability that he will win (a) only one egg is rotten. (a) exactly 3 events, (b) all the eggs are not rotten. (b) at least 4 events. 3. The probability that Michael Jackson will be late to 4. The probability that Tall Chen hits a target in an school is 0.15. Find the probability that in 5 days, archery competition is 0.6. If she attempts 8 times, the student will be late for calculate the probability that she will hit the target (a) exactly 2 days, (a) exactly 7 times, (b) not more than 1 day. (b) at least 7 times. 5. In a certain region, the probability that it will rain 6. It is known from a study that 70% of a type of seed on a day is 0.25. Calculate the probability that in a will germinate. If P.D.Vinoth has planted 10 seeds period of one week, it rains on in his garden, find the probability that (a) exactly 3 days, (a) exactly five seeds will germinate, (b) less than 3 days. (b) at least 8 seeds will germinate. 4 n r n-r Exercise 20.1.1 – Determine the Probability by Binomial Distributions P(X = r ) = Cr p q 7. Purple Chen walks to school everyday. The 8. Parkson Arwin will takes 10 subjects in the SPM probability that he will be late to school on any one examination. The probability that he will pass any day is 0.2. Find the probability that Purple Chen one of the subjects is 0.7. Find the probability that will be late on Parkson Arwin will pass (a) only two days, (a) all the 10 subjects, (b) more than 1 days. (b) at least 2 subjects. 9. The probability that a student will doze off during a 10. 25% of the teachers of a school have more than 2 lesson is 0.15. Find the number of students that must children be chosen so that the probability of finding at least (a) If 4 teachers are chosen at random from the one student doze off during a lesson exceed 99%. school, find the probability that only 3 teachers have more than 2 children. (b) Find the minimum number of teachers that must be chosen at random such that the chances of at least one teacher in the school having more than 2 children exceeds 90% 5 Exercise 20.1.2 – Plot the Graph of Binomial Distributions 1. In a chess competition, a player has to play 3 games. The probability a player to win is 0.6. Plot the graph of the binomial distribution. 2. The probability of David hitting the target in shooting competition is 0.45. Plot the graph of binomial distribution if David fires four shots. 6 Exercise 20.1.3 Mean, µ = np Variance, σ2 = npq Std. Dev., σ = npq Determine the Mean, the Variance and the Standard Deviation of Binomial Distribution 1. A fair dice is rolled 10 times continuously. The 2. The probability that a bomb which is released by a 1 jet fighter to hit its target is 0.8. If 40 bombs are probability of obtaining the number 6 is . Find 5 released by the jet fighter, determine the mean, the the mean, the variance and the standard deviation. variance and the standard deviation. 3. In a class of 32 students, 85% of them passed an 4. In an objective test, four options of answers are Additional Mathematics test. Find the mean, the given to a candidate to choose from. If the test variance and the standard deviation of the result of consists of 40 questions, find the mean, the the test. variance and the standard deviation of the test. 5. In the qualifying test for entrance to a university, 48 6. Given that three out of ten students in a school have candidates pass the test and the variance is 46.08. been selected to participate in a national service Find the number of candidate who sit for the test. program. If there are 270 students involved in the Hence, determine the probability of a candidate program, determine the number of students in the passing the test. school. Hence find its standard deviation. 7 20. 2 NORMAL DISTRIBUTION Important Notes 8 Exercise 20.2.1 – Sketch the Standard Normal Distribution Curve and Determine the Value of Probability 1. P(Z > 1.543) f(z) 2. P(Z < −0.852) f(z) z z 0 0 3. P(Z > 1.022) f(z) 4. P(Z ≤ 0.839) f(z) z z 0 0 5. P(Z ≥ − 0.35) f(z) 6. P(Z ≤ −0.635) f(z) z z 0 0 7. P(−0.75 < Z < 1.2) f(z) 8. P(0.5 < Z < 1.75) f(z) z z 0 0 9. P(−1.25 < Z < −0.6) f(z) 10. P(0 < Z < 1.228) f(z) z z 0 0 11. P(Z ≥ − 0.35) 12. P(−0.5 < Z < 1.25) 13. P(Z < −0.6) 14. P(−1.25 < Z < −0.6) 9 Exercise 20.2.2 – Sketch the Standard Normal Distribution Curve and Determine the z-value or k-value 1. P(Z > z) = 0.0883 2. P(Z < z) = 0.1401 0 0 3. P(Z > k ) = 0.2327 4. P(Z < k ) = 0.825) 0 0 5. P(Z > z ) = 0.75) 6. P(Z < z) = 0.5377 0 0 7. P(k < Z < 0) = 0.342 8. P(0 < Z < k) = 0.358 0 0 9. P(z < Z < 0.75) = 0.6 10. P(−0.5 < Z < z) = 0.54 0 0 11. P(Z > z) = 0.25 12. P(−2.21 < Z < k) = 0.9861 13. P(Z < k ) = 0.00641 14. P( z < Z < −0.65) = 0.2526 10 xμ z= Exercise 20.2.3 – Determine the z-value from a random chosen data and the probability σ 1. The masses of students in a school are normally 2. The height of students in a school are normally distributed with a mean of 52 kg and a standard distributed with a mean of 160 cm and variance of deviation of 2.5 kg. 144 cm2. (a) What at is the z-value the mass of a student (a) Determine the z-value of the height if a student chosen at random has a mass of 50 kg. chosen at random has a height of less than 162 (b) Hence find the probability of chosen such a cm. student (b) Hence, the probability of chosen a student of such height. 3. The score of a class for a monthly test are normally 4. The scores of Add. Maths test for a class are distributed with a mean of 48 marks and a standard normally distributed with a mean of 55 marks and a deviation of 15 marks. Determine the z-value for the standard deviation of 16. Determine marks between 40 marks and 65 marks. Hence, the (a) the z-value of the scores if a student is chosen probability of chosen a student with such the range with the marks less than 40. of scores. (b) the probability of chosen such a student. 5. The volume of bottled fruit juice produced by a 6. X is a random variable of a normal distribution with factory are normally distributed with a mean of 750 a mean of 34 and a variance of 9. Find ml and a standard deviation of 12 ml. If a bottle of (a) the z-value for X is between 34 and 37. the fruit juice is chosen at random, find, (b) the probability for 31<X<37. (a) the z-value of the volume of the fruit is less than 740 ml, (b) the probability of chosen such the volume of the fruit juice. 11 20.3 PAST YEAR QUESTIONS Exercise 20.3.1 – SPM Paper 1 SPM 2005: SPM 2007: The mass of students in a school has a normal X is continuous random variable of a normal distribution with a mean of 54 kg and a standard distribution with a mean of 52 and a standard deviation of 12 kg. Find deviation of 10. Find (a) the mass of the students which gives a standard (a) the z-score when X = 67.2 score of 0.5, (b) the value of k when P(z < k) = 0.8849 (b) the percentage of students with mass greater than 48 kg. [4 marks] [ 4 marks] SPM 2006: SPM 2007 Diagram 10 shows a standard normal distribution graph. The probability that each shot fired by Ramli hits a The probability represented f(z) target is 1 . by the area of the shaded 3 region is 0.3485. 0.3485 (a) If Ramli fires 10 shots, find the probability (a) Find the value of k. z that exactly 2 shaots hit the target. 0 k (b) X is a continuous random (b) If Ramli fires n shots, the probability that 1 variable is normally Diagram 10 all the n shots hit the target is . [4 marks] 243 distributed with a mean of 79 and a standard Find the value of n. deviation of 3. Find the value of X when the z-score is k [4 marks] 12 SPM2008: SPM 2009: The masses of students in a school have a normal The masses of apples in a stall have a normal distribution with a mean of 40 kg and a standard distribution with a mean of 200 g and a standard deviation of 5 kg. Calculate the probability that a deviation of 30 g. student chosen at random from this group has a (a) Find the mass, in g, of an apple whose z-score mass of is 0.5. (a) more than 45 kg, (b) between 35 kg and 47.8 kg. [4 marks] (b) If an apple is chosen at random, find the probability that the apple has a mass of at least 194 g. [ 4 marks] SPM 2010: SPM 2004: The discrete random variable X has a binomial X is a random variable of a normal distribution with a probability distribution with n = 4, where n is the mean of 5.2 and a variance of 1.44. Find number of trials. Diagram 25 shows the probability (a) the z –value if X = 6.7 distribution of X. (b) P(5.2 ≤ X ≤ 6.7). [4 marks] P( X = x) k 1 4 Diagram 25 1 16 x 0 1 2 3 4 Find (a) the value of k. (b) P(X ≥ 3). [4 marks] 13 Exercise 20.3.2 – SPM Paper 2 SPM 2005: (a) The result of a study shows that 20% of the pupils in a city cycle to school. If 8 pupils from the city are chosen at random, calculate the probability that (a) exactly 2 of them cycle to school. (b) less than 3 of them cycle to school. [4 marks] (b) The masses of water melons produced from an orchard follow a normal distribution with a mean of 3.2 kg and a standard deviation of 0.5 kg. Find (i) the probability that the watermelon chosen randomly from the orchard has a mass of not more than 40 kg. (ii) the value of m if 60 % of the watermelons from the orchard have a mass of more than m kg. [6 marks] 14 SPM 2006: An orchard produces lemon. Only lemons with diameter , x greater than k cm are graded and marketed. Table 3 shows the grades of the lemons based on their diameters. C Grade A B Diameter, x (cm) x>7 7≥x>5 5≥ x > k Table 3 It is given that diameters of the lemons have a normal distribution with a mean of 5.8 cm and a standard deviation of 1.5 cm. (a) If one lemon is picked at random, calculate the probability that it is of grade A. [2 marks] (b) In a basket of 500 lemons, estimate the number of grade B lemons. [2 marks] (c) If 85.7% of the lemons is marketed, find the value of k. [4 marks] 15 SPM 2007: (a) In a survey carried out in a school, it is found that 2 out 5 students have handphones. If 8 students from that school are chosen at random, calculate the probability that (i) exactly 2 students have handphones, i. (ii) more than 2 students have handphones. [5 marks] (b) A group of workers are given medical check up. The blood pressure of a worker has a normal distribution with a man of 130 mmHg and a standard deviation of 16 mmHg. Blood pressure that is more than 150 mmHg is classified as ‘high blood pressure’. (i) A worker is chosen at random from the group. Find the probability that the worker has a blood pressure between 114 mmHg and 150 mmHg.more than 40 kg. (ii) It is found that 132 workers have ‘high blood pressure’. Find the total number of workers in the group. [5 marks] 16 SPM 2008: The masses of mangoes from an orchard have a normal distribution with a mean of 300 g and a standard deviation 0f 80 g. (a) Find the probability that a mango chosen randomly from this orchard has a mass of more than 168 g. [3 marks] (b) A random sample of 500 mangoes is chosen. (i) Calculate the number of mangoes from this sample that have a mass of more than 168 g. (ii) Given that 435 mangoes from this sample have a mass of more than m g, find the value of m. [7 marks] 17 SPM 2009: A test paper consists of 40 questions. Each question is followed by four choices of answer, where only one of these is correct. (a) Salma answers all the questions by randomly choosing an answer for each question. (i) Estimate the number of questions she answered correctly. i. (ii) Find the standard deviation of the number of questions she answered correctly. [5 marks] (b) Basri answers 30 questions correctly and randomly chooses an answer for each of the remaining 10 questions. Find the probability that he answers. (i) 36 questions correctly, (ii) at least 32 questions correctly. [5 marks] 18 SPM 2010: (a) A random variable, X, has a binomial distribution with 10 trials where the probability of success in each trial is p. The mean number of success is 4. calculate (i) the value of p, ii. (ii) P(X ≤ 2) [5 marks] (b) The diameters of limes from a farm have a normal distribution with a mean of 3.2 cm and a standard deviation of 1.5 cm. Calculate (i) the probability that a lime chosen at random from this farm has a diameter of more than 3.9 cm. (ii) the value of d if 33% of the limes have diameters less than d cm. [5 marks] 19 SPM 2004: (a) A club organizes a practice session for trainees on scoring goals from penalty kicks. Each trainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick is p. After the session, it is found that the mean number of goals for a trainee is 4.8. (i) Find the value of p. iii. (ii) If a trainee is chosenat random, find the probability that he scores at least one goal. [5 marks] (b) A survey on body-mass is done on a group of students. The mass of a student has a normal distribution with a mean of 50 kg and a standard deviation of 15 kg. (i) If a student is chosen at random, calculate the probability that his mass is less than 41 kg. (ii) Given that 12% of the students have a mass of more than m kg, find the value of m. [5 marks] 20 SPM 2003: (a) Senior citizens make up 20% of the population of a settlement. (i) If 7 people are randomly selected from the settlement, find the probability that at least two of them are senior citizens. iv. (ii) If the variance of the senior citizens is 128, what is the populations of the settlement? [5 marks] (b) The mass of the workers in a factory is normally distributed with a mean of 67.86 kg and a variance of 42.25 kg2. 200 of the workers in the factory weigh between 50 kg and 70 kg. Find the total number of workers in the factory. [5 marks] 21 THE END