22 S2 Probability APQ (RVHS)

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 12 Additional Practice Questions 1. [HCI/2007/JC2/CT] Loy is a member of his school’s soccer team, and the team is participating in an invitation tournament where each team is to play a total of 3 matches. The probabilities that Loy gets selected to play in the first match and the second match are and respectively. The probability that Loy gets selected to play in the third match, given that he was selected to play in the second match is ; and the probability that Loy gets selected to play in the third match, given that he was not selected to play in the second match is . Find the probability that (i) Loy is not selected to play in the third match, (ii) Loy gets selected to play in all the 3 matches, (iii) Loy gets selected to play in at most 1 of the 3 matches. [(i)଻ ଵହ (ii) ସ ଶ଻ (iii) ଶଷଷ ଽସହ] Let event S = Loy gets selected to play (i) P(2nd S, 3rd S’) + P(2nd S’, 3rd S’) = (4/7)(2/3) + (3/7)(1/5) = 7/15 (ii) (7/9)(4/7)(1/3) = 4/27 (iii) P(1st S’, 2nd S’, 3rd S’) + P(S, S’, S’) + P(S’, S, S’) + P(S’, S’, S) = (2/9)(3/7)(1/5) + (7/9)(3/7)(1/5) + (2/9)(4/7)(2/3) + (2/9)(3/7)(4/5) = 233/945 7 9 4 7 1 3 4 5 S S’ S S’ S S’ 7/9 4/7 2/9 3/7 4/7 3/7 S S’ S S’ 1/3 2/3 4/5 1/5 S S’ S S’ 1/3 2/3 4/5 1/5 1st match 2nd match 3rd match

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 13 2. [ACJC/2010/Prelim/8] A girl is given two toys, a doll and a teddy bear. The probability that she plays with the doll is 0.5 and the probability that she plays with the teddy bear is 0.6. The probability that she plays with the doll or the teddy bear or both is 0.8. Find the probability that (i) She plays with the doll but not the teddy bear. (ii) She plays with the doll given that she does not play with the teddy bear. [(i) 0.2; (ii) 0.5]

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 14 3. [RI(JC)/2011/Promo/9] A group of students take a physical fitness test. A student who fails the test on the first attempt is allowed one further attempt. For a randomly chosen student, the probability of passing the test on the first attempt is 0.6 and the probability of passing on the second attempt is 0.8. (i) Draw a tree diagram to represent the possible outcomes. (ii) Find the probability that a randomly chosen student fails the test on both attempts. (iii) Given that a student passes the test, find the probability that it is on the second attempt. (iv) Three students taking the test are chosen at random. Find the probability that two of them pass on the first attempt and the other passes on the second attempt. [(ii) 0.08; (iii) 0.348; (iv) 0.3456]

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 15 4. [AJC/2011/Promo/10] (a) E and F are two events such that , and . Find p if (i) E and F are mutually exclusive; (ii) E and F are independent; (iii) E is a subset of F. (b) The following information is known about two events X and Y. , and . Find the following probabilities: (i) (ii) (iii) [(a) (i) 1 12; (ii) 1 9; (iii) 1 3; (b) (i) 7 25; (ii) 53 100; (iii) 7 20 ]  1P 4E  1P 3E F P F p  2P 5X  7P 10Y X 1P 4X Y  P X Y P Y P X Y 

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 16 5. [CJC/2019/Prelim/02/Q11] A palindrome is a string of letters or digits that is the same when you read it forwards or backwards. For example: HHCCHH, RACECAR, STATS are palindromes. A computer is instructed to use any of the letters R, O, F, L to randomly generate a string of 5 letters. Repetition of any letter is allowed, but the string cannot contain only one letter. For example, RRRRR is not allowed. Events A and B are defined as follows: A: the string generated contains 2 distinct letters B: the string generated is a palindrome (i) Find P A and P B . [5] (ii) Find P A B and hence determine if A and B are independent. [3] (iii) Find the probability that the string generated either contains 2 distinct letters, or that it is a palindrome, or both. [2] (iv) Find the probability that the string generated contains 2 distinct letters, given that it is a palindrome. [2] [(i)   3 1P , P17 17A B  (ii) 3 85 (iii) 1 5 (iv) 3 5] Solution (i) No of ways to form the string unrestricted = 54 4 1020  No of ways to form the string using 2 letters = 4 5 2C 2( 2)  = 180 OR 4 4 2 2 5! 5!2 2 1804! 2!3!C C      Hence 180 3P( ) 1020 17A   = 0.176 No of ways to have a palindrome = 34 4 = 60 OR 4 4 2 3 3!2 3! 60! C2C     Hence 60 1P( ) 1020 17B   = 0.0588 (ii) No of ways to form palindrome of 2 letters = 4 3 4 2 2C 2 2) or 6 3!( 2C 2 3!   Hence 36 3P( ) 0.03531020 85A B    P( ) P( ) 0.0104A B  Since P( ) P( ) P( )A B A B , A and B are not independent. (iii) 3 1 3 1P( ) P( ) P( ) P( ) 0.2 17 17 85 5A B A B A B        (iv)     3P 385P | 0.6 1P 5 17 A BA B B   

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 17 6. [HCI/2020/Prelim/02/Q7] [(a) 6 7 (b) 95 156 (c) 4 15] 7(a) Required probability 13 2 13! 2!13 15! 15 C   6 7 or 0.857 (3 s.f.) Alternative Total numbers of ways = 15 1 ! 14!  Number of ways leaders seated together 14 1 ! 2 13! 2     Probability = 13! 2 1 61 114! 7 7     7(b) Let X be the event that the 3 students from the same class are seated separately. Let Y be the event that both leaders are seated separately. Required probability P P( ) X Y Y  A team of 15 students was selected for an outdoor education trip. One student volunteered to be the trip leader while another volunteered as the assistant trip leader. They decided to have some ice-breaker games, where all 15 students sat in a circle. (a) Find the probability that both leaders were not seated together. [2] During the ice-breaker games, it was realised that 3 of the other 13 students belonged to the same class. (b) Find the probability that none of these three students sat next to each other, given that both leaders were not seated together. [3] (c) After the ice-breaker games, both leaders decided to randomly break the team up into two groups to discuss about administrative matters and training program for the trip. The group which discussed about the training program consisted of 7 students. Find the probability that the trip leader was heading the training program group while the assistant leader was in charge of the administrative matters group. [2]

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 18   P P 'X X Y P Y   12 11 3 3 12! 11! 3! 2! 3!12 11 15! 1567 C C       95 156 or 0.609 ( 3 s.f.) 7(c) Required probability 13 6 15 7 C C 4 15 or 0.267 (3 s.f) 7. [VJC/2016/Prelim/7] Box A contains 10 red, 8 blue and 7 green balls. Box B contains 2 white and 3 black balls. All the balls are indistinguishable except for their colours. Three balls are taken from Box A and two balls are taken from Box B, at