Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES


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1 CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times and adequate recording of the happening of event.. In a theoretical approach to probability, we try to predict what will happen without actually performing the experiment. 3. An outcome of a random experiment is called an elementary event. 4. The theoretical (classical) probability of an event E, written as P(E), is defined as Number of outcomes favourable to E P(E)= Number of all possible outcomes of the experiment 5. The probability of an impossible event is 0, and that of sure event is The probability of an event E is a real number P(E) such that 0 P(E) An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is For any event E, P(E) + P( E ) = 1, where E stands for not E. 9. Total possible outcomes, when a coin is tossed n times, is n. 10. Total possible outcomes, when a die is thrown n times, is 6 n. 11. Playing Cards (Total 5) Spade ( ) Club ( ) Heart ( ) Diamond ( ) (Black coloured) (Black coloured) (Red coloured) (Red coloured) The cards in each suit are ace, king, queen, jack and number cards to 10. Kings, queens and jacks are called face cards. ILLUSTRATIVE EXAMPLES Example 1. An unbiased die is thrown. What is the probability of getting : (i) an odd number a multiple of 3 (iii) a perfect square number (iv) a number less than 4. Solution. Here, total number of all possible outcomes= 6 (i) favourable outcomes are 1, 3, 5. So, no. of favourable outcomes = 3 P (an odd number) No. of favourable outcomes 3 = 1 Total no. of possible outcome 6 56 PROBABILITY MATHEMATICS X
2 favourable outcomes are 3 and 6. So, no. of favourable outcomes = P (a multiple of 3) (iii) favourable outcomes are 1 and 4. So, no. of favourable outcomes = P (a perfect square number) (iv) favourable outcomes are 1, and 3. So, no. of favourable outcomes = 3 3 P (a number less than 4) 1 6 Example. Three unbiased coins are tossed together. Find the probability of getting : Solution. Example 3. Solution. (i) all heads two heads (iii) one head (iv) at least two heads Here, possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH and TTT. So, total no. of possible outcomes = 8 (i) favourable outcome = HHH So, No. of favourable outcome = 1 no. of favourable outcomes P (all heads) 1 Total no. possible outcomes 8 favourable outcomes are HHT, THH and HTH. So, no. of favourable outcomes = 3 P (two heads) 3 8 (iii) favourable outcomes are HTT, THT and TTH. So, no. of favourable outcomes = 3 P (one head) 3 8 (iv) favourable outcomes are HHH, HHT, HTH and THH. So, no. of favourable outcomes = 4 4 P (at least two heads) 1 8. Find the probability that a leap year selected at random will contain 53 sundays. In a leap year, there are 366 days. We have, 366 days = 5 weeks + days. Thus, a leap year has always 5 sundays. MATHEMATICS X PROBABILITY 57
3 The remaining days can be : (i) Sunday and Monday Monday and Tuesday (iii) Tuesday and Wednesday (iv) (v) (vi) Wednesday and Thursday Thursday and Friday Friday and Saturday (vii) Saturday and Sunday Clearly, there are seven elementary events associated with this random experiment. Let E be the event that a leap year has 53 sundays. Clearly, the event E will happer if the last two days of the leap year are either Sunday and Monday or Saturday and Sunday. Favourable no. of elementary events = Hence, required probability Ans. 7 Example 4. One card is drawn from a pack of 5 cards, each of the 5 cards being equally likely to be drawn. Find the probability that the card drawn is : (i) an ace either red or king (iii) a face card Solution. here, total no. of possible outcomes = 5. (i) (iv) a red face card There are 4 ace cards in a pack of 5 cards. One ace can be chosen in 4 ways. So, favourable no. of outcomes = 4 no. of favourable outcomes P (an ace) = Total no. of possible outcomes 4 = 1 5 There are 6 red cards, including red kings. Also, there are 4 kings, two red and two black. card drawn will be a red card or a king if it is any one of 8 cards (6 red cards and black kings) So, favourable no. of outcomes = 8 P(either red or king) (iii) Kings, queens and jacks are the face cards. So, favourable no. of outcomes = 3 4 = 1 (iv) 1 P(a face card) 3 5 There are 6 red face cards, 3 each from diamonds and hearts. So, favourable no. of outcomes = 6 P(a red face card) PROBABILITY MATHEMATICS X
4 Example 5. Solution. Example 6. Cards marked with the numbers to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number of the card is : (i) an even number a number less than 14 (iii) a number which is a perfect square (iv) a prime number less than 0. From to 101, these are (101 ) + 1 = 100 numbers. So, total no. of possible outcomes = 100. (i) From to 101, the even numbers are, 4, 6,..., 100 which are 50 in number. So, number of favourable outcomes = 50 no. of favourable outcomes P(an even number) = Total no. of possible outcomes 50 = From to 101, the numbers less than 14 are, 3,..., which are 1 in number. So, no. of favourable outcomes = 1 1 P(a number less than 14) (iii) From to 101, the perfect squares are 4, 9, 16, , which are 9 in number. So, no. of favourable outcomes = 9 P (a number which is a perfect square) (iv) From to 101, the prime numbers less than 0 are, 3, 5, 7, 11,, 17 and 19 which are 8 in number. So, no. of favourable outcomes = 8 P (a prime no. less than 0) A bag contains 3 red balls and 5 black balls.a ball is drawn at random from a bag. What is the probability that the ball drawn is : (i) red not red [NCERT] Solution. Total number of balls = = 8 Example 7. (i) no. of red balls 3 P (red ball) = = Total no.of balls 8 3 P (not red ball) = 1 P(red ball) A jar contains 4 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is. Find the number of blue marbles in the jar. 3 [NCERT] MATHEMATICS X PROBABILITY 59
5 Solution. Total number of elementary events = 4. Let there be x green marbles. P (green marbles is drawn) = 4 x but, P(green marbles is drawn) 3 x x 4 x Number of green marbles = 16 (given) Number of blue marbles = 4 16 = 8 Ans. Example 8. Suppose you drop a die at random on the rectangular region shown in the figure. What is the probability that it will land inside the circle with diameter 1 m? [NCERT] 3m Solution. Total area of rectangular region = 3 m m = 6 m 1 Area of the circle r m m 4 P (die to land inside the circle) / 4 = π 6 4 Ans. Example 9. A bag contains 1 balls out of which x are white. (i) If one ball is drawn at random, what is the probability that it will be a white ball? 60 PROBABILITY MATHEMATICS X m If 6 more white balls are put in the bag, the probability of drawing a white ball will be double than that in (i). Find x. [NCERT] Solution. (i) Total number elementary events = 1. There are x white balls out of which one can be chosen in x ways. So, favourable number of elementary events = x no. of favourable outcomes p 1 = P (white ball) = Total no. of possible outcomes = 1 x If 6 more white balls are put in the bag, then total number of balls in the bag =1 + 6 = 18 and, no. of white balls in the bag = x + 6 x 6 p P (getting a white ball) 18 It is given that, p p1 x 6 x 18 1
6 x 6 x 6( x 6) 18x x + 36 = 18x 1 x = 36 x = 3 Ans. Example 10. Two dice are thrown simultaneously. Find the probability of getting : (i) a doublet i.e. same number on both dice. the sum as a prime number. Solution. Possible outcomes associated to the random experiment of throwing two dice are : (1, 1), (1, ),..., (1, 6) (, 1), (, ),..., (, 6) (6,1), (6, ),..., (6, 6) Total number of possible outcomes = 6 6 = 36 (i) The favourable outcomes are (1, 1), (, ), (3, 3), (4, 4), (5, 5) and (6, 6). Total no. of favourable outcomes = 6 So, P(a doublet) = no. of favourable outcomes Total no. of possible outcomes 6 = Here, favourable sum (as a prime number) are, 3, 5, 7 and 11. So, favourable outcomes are (1, 1), (1, ), (, 1), (1, 4), (4, 1), (, 3), (3, ), (1, 6), (6, 1), (, 5), (5, ), (3, 4), (4,3), (6, 5) and (5, 6). no. of favourable outcomes = 15 P (the sum as a prime number) 15 5 Ans PRACTICE EXERCISE 1. The probability that it will rain tomorrow is What is the probability that it will not rain tomorrow?. A die is thrown once. Find the probability of gething : (i) a multiple of 3 a multiple of or 3 (iii) a prime number 3. Two coins are tossed simultaneously. Find the probability of getting : (i) two heads exactly one tail (iii) no tail 4. Three unbiased coins are tossed simultaneously. Find the probability of getting : (i) one head two heads (iii) All heads (iv) at least two heads (v) at least one head and one tail 5. A die is thrown once. What is the probability of getting : (i) an even number an odd number (iii) a number 3 (iv) a number 5 or 6 (v) a number > 6 6. In a simultaneous throw of a pair of dice, find the probability of getting : (i) 7 as a sum a doublet of odd numbers (iii) not a doublet (iv) an odd number on the first die MATHEMATICS X PROBABILITY 61
7 (v) a sum less than 6 (vi) a sum more than 10 (vii) neither 9 nor 11 as the sum of the numbers on the faces (viii) a total of atleast 10 (ix) a multiple of 3 as the sum (x) a doublet of prime numbers 7. What is the probability that an ordinary year has 53 mondays? 8. A card is drawn at random from a pack of 5 cards. Find the probability that the card drawn is : (i) a jack, queen or a king a face card (iii) a black king (iv) black and a king (v) neither a heart nor a king (vi) spade or an ace (vii) a queen of diamond (viii) either a black card or a king (ix) an ace of heart (x) neither an ace nor a king (xi) 10 of black suit (xii) a club (xiii) neither a red card nor a queen (xiv) a 8 of heart (xv) an ace of red colour 9. Two black kings and two black jacks are removed from a pack of 5 cards. Find the probability of getting: (i) a card of hearts a card of clubs (iii) a king (iv) a black card (v) either a red card or a king (vi) a red king (vii) neither an ace nor a king (viii) a jack, queen or a king 10. A bag contains 4 red balls, 5 black balls and 3 green balls. A ball is drawn at random from a bag. Find the probability that the ball drawn is : (i) a red ball a black ball (iii) not a green ball (iv) a black or a green ball 11. A bag contains 5 red balls and some black balls. If the probability of drawing a black ball is double that of a red ball, find the number of black balls in the bag. 1. A bag contains 5 red, 8 white and 7 black balls. A ball is drawn at random from the bag. Find the probability that the ball drawn is ; (i) red or white not black (iii) neither white nor black [CBSE 005]. 17 cards numbered 1,, 3,..., 17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is : (i) odd a prime (iii) divisible by 3 (iv) divisible by and 3 both 14. Balls marked with numbers to 101 are placed in a box and mixed thoroughly. One ball is drawn at random from this box. Find the probability that the number on the ball is : (i) an even number an odd number (iii) a number less than 17 (iv) a number which is a perfect square (v) a number which is a perfect cube (vi) a number divisible by 9 (vii) a prime number less than 41 (viii) a number which is divisible by 3 or Find the probability that a number selected from the number 1 to 5 is not a prime number when each of the given numbers is equally likely to be selected. [CBSE 005] 6 PROBABILITY MATHEMATICS X
8 16. Find the probability that a number selected at a random from the numbers 1,, 3,..., 35 is a (i) prime number multiple of 7 [CBSE 006 (C)] (iii) a multiple of 3 or A bag contains 3 red, 5 black and 7 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is : (i) white red (iii) not black (iv) red or white [CBSE 004] 18. Out of 400 bulbs in a box, 15 bulbs are defective. One bulb is taken out at random from the box. Find the probability that the drawn bulb is not defective. [CBSE 004(C)] 19. A card is drawn from a well shuffled pack of 5 cards. Find the probability that the card is neither a red card nor a queen. [CBSE 005] 0. A bag contains 5 white balls, 7 red balls, 4 black balls and blue balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is : (i) white or blue red or black (iii) not white (iv) neither white nor black [CBSE 006] 1. A card is drawn at random from a wellshuffled deck of playing cards. Find the probability that the card drawn is : (i) a card of spade or an ace a red king (iii) neither a king nor a queen (iv) either a king or a queen [CBSE 006]. Cards marked with numbers 3, 4, 5,..., 50 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that number on the card drawn is : (i) divisible by 7 a number which is a perfect square [CBSE 007] 3. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is four times that of a red ball, find the number of blue balls in the bag. [CBSE 007] tickets of a lottery were sold and there are 5 prizes on these tickets. If Saket has purchased one lottery ticket, what is the probability of winning a prize? 5. A child has a block in the shape of a cube with one letter written on each face as shown below : A B C D E A The cube is thrown once. What is the probability of getting : (i) A? D? 6. Two customers are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit on any one day as on another. What is the probability that both will visit the shop on : (i) the same day different days? (iii) Consecutive days 7. The letters A, B, C, A, D, E, B, A, B and F are marked on different cards such that one card has only one letter on it. A man draws one card. Find the probability that the card drawn is marked : (i) letter A letter B (iii) letter B or D (iv) letter A or C 8. A bag contains 36 balls out of which x are black. (i) If one ball is drawn at random, what is the probability of getting a black ball? If 1 more black balls are put in the bag, the probability of drawing a black ball is 1. Find x. MATHEMATICS X PROBABILITY 63
9 9. From a pack of 5 playing cards jacks, queens, kings and aces of red colour are removed. From the remaining, a card is drawn at random. Find the probability that the card drawn is : (i) a black queen a red card (iii) a black jack (iv) a picture card (Jacks, queens and kings are picture cards) [CBSE 006 (C)] 30. A letter is chosen at random from the letters of the word MATHEMATICS. Find the probability that the letter chosen is a (i) vowel consonant. 31. Ram and Shyam are friends. What is the probability that both will have : (i) different birthday? the same birthday? (ignore the leap year). 3. In a family, there are 3 children. Assuming that the chances of a child being a male or a female are equal, find the probability that : (i) there is one girl in the family. there is no male child in the family. (iii) there is atleast one male child in the family. 33. A number x is selected from the numbers 1,, 3 and then a second number y is selected randomly from the numbers 1, 4, 9. What is the probability that the product xy < 9? 34. A number is selected randomly from all possible 3digit numbers. What is the probability that the number selected is : (i) an odd number having all 3 digits same (iii) divisible by If x and y are natural numbers such that 1 x 4 and 3 y 6. What is the probability that : (i) x y 8 xy is even. HINTS TO SELECTED QUESTIONS 6. Here, total no. of possible outcomes = 36 favourable outcomes are (1, 1), (3, 3) and (5, 5) (iii) P (not a doublet) = 1 P (a doublet) (iv) favourable outcomes are (1, 1), (1, ),..., (1, 6); (3, 1), (3, ),..., (3, 6); (5, 1), (5, ),..., (5, 6) which are total 18 cases. (v) P (a sum less than 6) = P (a sum =, 3, 4 or 5) (vi) P (a sum more than 10) = P (a sum = 11 or 1) (vii) P (neither 9 nor 11 as the sum) = 1 P (either 9 or 11 as the sum) (viii) P (a total of atleast 10) = P (a total of 10 or 11 or 1) 8. Total number of possible outcomes = 5. (iv) P (black and a king) as there are black kings. 5 (v) P (neither a heart nor a king) = 1 P (either a heart or a king) 6. Possible outcomes are : TT TW TTh TF TS WT WW WTh WF WS ThT ThW ThTh ThF ThS FT FW FTh FF FS ST SW STh SF SS So, total no. of possible outcomes are PROBABILITY MATHEMATICS X
10 (i) P (both visit on the same day) 5 1 ( Possible outcomes are TT, WW, ThTh, FF, SS) 5 5 P (both visit on different days) = 1 P (both visit on same day) (iii) P (both visit on consecutive days) 8 5 ( Possible outcomes are ThW, WT, WTh, TW, ThF, FTh, FS, SF) 31. Each may have birthday on any day (365 days) of the year. Thus, the possible outcomes are 365. (i) If Shyam s birthday is different from Ram s, the number of favourable outcomes for him is = 364. Required probability = 1 P (both have different birthdays). 3. In a family, the three children could be as follows : BBB, BBG, BGG, GGG i.e. in all 4 possible outcomes. 33. Two numbers can be selected in following 9 ways : (1, 1), (1, 4), (1, 9), (, 1), (, 4), (, 9), (3, 1), (3, 4), (3, 9). here, favourable outcomes are (1, 1), (1, 4), (, 1), (, 4), (3, 1). 34. Total 3 digit numbers are 900. (i) clearly, possible outcomes = 450. here, possible outcomes are 111,,..., 999. (iii) possible outcomes are 10, 105, 108,..., 999 which are 300 in numbers. 35. Total possible outcomes are 4 4 = 16, which are (1, 3), (1, 4), (1, 5), (1, 6), (, 3), (, 4), (, 5), (, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 3), (4, 4), (4, 5), (4, 6) (i) favourable outcomes are (, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6). favourable outcomes are (1, 4), (1, 6), (, 3) (, 4), (, 5), (, 6), (3, 4), (3, 6), (4, 3), (4, 4), (4, 5), (4, 6). MULTIPLE CHOICE QUESTIONS Mark the correct alternative in each of the following : 1. In a single throw of a die, the probability of getting a multiple of 3 is : (a) 1 (b) 1 (c) Which of the following cannot be the probability of an event? (a) 3 (b) 0.4 (c) 35% (d) If P(A) = 0.04, then P (Not A) is : (a) 0.06 (b) 0.04 (c) 0.9 (d) The probability that a number selected at random from the number 1,,, 3, 3, 3, 4, 4, 4, 4 will be their average is (a) 5 (b) 3 5 (c) 3 10 MATHEMATICS X PROBABILITY 65 (d) 3 (d) none of these
11 5. Pranshi and Ria are friends. The probability that both will have same birthday (ignoring a leap year) is: (a) (b) (c) (365) 6. Two dice are thrown once. The probability of getting a total of 7 or 9 is : (a) 1 (b) (c) Three coins a tossed once. The probability of getting at least heads is : (d) none of these (d) 7 18 (a) 1 (b) 3 (c) 3 (d) none of these A card is drawn at random from a well shuffled deck of 5 cards. The probability of getting a face card of black colour is : (a) 1 (b) 1 (c) 3 (d) There are 30 cards, of same size, in a bag on which numbers 1 to 30 are written. Once card is taken out of the bag at random. The probability that the number on the selected card is not divisible by 3, is : (a) 1 (b) 3 (c) (d) A bag contains 5 white balls, 3 black balls and 4 white balls. A ball is drawn out of the bag at random. The probability that the ball is red or white is : (a) 1 (b) PROBABILITY MATHEMATICS X (c) 1 4 (d) none of these VERY SHORT ANSWER TYPE QUESTIONS (1 MARK QUESTIONS) 1. A coin is tossed twice. Write the possible outcomes.. A die is rolled twice. Write the possible outcomes. 3. If the probability of winning a game is 0.4. What is the probability of losing it? 4. A die is thrown once. What is the probability of getting a prime number? 5. Two coins are tossed simultaneously. What is the probability of getting atleast one head? 6. A bag contains 6 red and 4 blue balls. One ball is drawn at random. What is the probability that the ball drawn is not red? 7. A card is drawn at random from a pack of 5 playing cards. What is the probability that the card drawn is neither a red card nor a black king? 8. A box contains 30 cards, numbered from 1 to 30. One card is drawn at random from the box. What is the probability that the number on the drawn card is a multiple of or 3? 9. Out of 00 bulbs in a box, 0 bulbs are defective. One bulb is taken out at random from the box. What is the probability that the drawn bulb is not defective? 10. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is four times that of a red ball, how many blue balls are in the bag? 11. Three coins are tossed simultaneously. What is the probability of getting no head? 1. Two dice are thrown simultaneously. What is the probability of getting a doublet?
12 . A letter is chosen from the word RANDOM. What is the probability that it is a vowel? 14. Two dice are thrown simultaneously. What is the probability of getting a multiple of 3 as the sum? 15. A child has a die whose six faces show the letters as given below : A B B C D E The die is throw once. What is the probability of getting B? PRACTICE TEST M.M : 30 General Instructions : Qs. 14 each carries marks, Qs. 58 each carries 3 marks and Qs each carries 5 marks. Time : 1 hour 1. A box contains 100 bulbs out of which 10 are defective. What is the probability that if a bulb is drawn, it is (i) defective nondefective.. An integer is chosen from 1 to 15. Find the probability that the integer chosen is divisible by A bag contains 8 red, 6 white and 4 black balls. A ball is drawn at random from the bag. Find the probability that the drawn ball is : (i) red or white neither white nor black. 4. From a well shuffled pack of 5 cards, two black kings and two black jacks are removed. From the remaining cards, a card is drawn at random. Find the probability that drawn card is neither an ace nor a king. 5. A die, in the shape of a tetrahedron, has four faces on which numerals 3, 4, 5, 8 are written. The die is rolled. Find the probability of getting an even number. 6. A bag contains 5 red balls and some black balls. If the probability of drawing a black ball is double that of a red ball, find the number of black balls in a bag. 7. A pair of dice is thrown once. Find the probability of getting the sum of numbers on two dice as A piggy bank contains hundred 50 p coins, fifty Re 1 coins, twenty Rs. coins and ten Rs. 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin : (i) will be a 50 p coin? will not be a Rs. 5 coin? 9. Cards marked with numbers 3 to 15 are thoroughly mixed. If one card is drawn at random; find the probability that the number on the card is : (i) an odd number a number less than 5 (iii) a number greater than 140 (v) a prime number between 10 and 40. (iv) a number which is a perfect square 10. A card is drawn at random from a wellshuffled deck of playing cards. Find the probability that the card drawn is : (i) a card of spade or an ace a red king (iii) neither a king nor a queen (iv) either a king or a queen (v) a face card MATHEMATICS X PROBABILITY 67
13 ANSWERS OF PRACTICE EXERCISE (i) (iii) 1 3. (i) (iii) 1 4 (iv) 3 4 (v) (i) (iii) 1 8 (iv) 1 (v) (i) 1 1 (iii) 3 (iv) 1 3 (v) 0 6. (i) (iii) 5 6 (iv) 1 (v) 5 18 (vi) 1 1 (vii) 5 6 (viii) 1 6 (ix) 1 3 (x) (i) 3 3 (iii) 1 6 (iv) 1 6 (v) 9 (vi) 4 (vii) 1 5 (viii) 7 (ix) 1 5 (xi) 1 6 (xii) 1 4 (xiii) 6 (xiv) 1 5 (xv) (i) (iii) 1 11 (iv) (v) (vi) 1 4 (vii) 7 8 (viii) (i) (iii) 3 4 (iv) 3. (i) (iii) 5 17 (iv) 17 (viii) (i) (iii) (i) (iii) 6 (iv) 5. (i) (i) 36 x 30. (i) (i) (i) (x) (iii) (i) 1 1 (iii) 3 0 (iv) (v) (vi) (vii) (i) (iii) (i) (iii) 3 (iv) 3 0 (iii) 5 (iv) (i) (i) 1 9 (iii) 1 (iv) (i) (i) (iii) (i) (iii) 5 3. (i) (iii) (i) PROBABILITY MATHEMATICS X
14 ANSWERS OF MULTIPLE CHOICE QUESTIONS 1. (b). (b) 3. (d) 4. (c) 5. (a) 6. (c) 7. (a) 8. (c) 9. (c) 10. (a) 1. {HH, HT, TH, TT} ANSWERS OF VERY SHORT ANSWER TYPE QUESTIONS. {(1, 1), (1, ),..., (1, 6), (, 1),..., (, 6),..., (6, 1),..., (6, 6)}; Total 36 cases (i) (i) (iii) (i) ANSWERS OF PRACTICE TEST (i) (iv) 150 (v) (iii) (iv) (v) (i) MATHEMATICS X PROBABILITY 69
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