RI S1B Probability tutorial (Solns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _____________________ Tutorial S1B: Probability Page 1 of 16 Tutorial S1B: Probability 1 It is given that events A and B are independent and 5P( ) 8AB , 7P( ') 24AB . Calculate (i) P( )B , (ii) P( )A , (iii) P( )AB , (iv) P( ' ')AB . [(i) 1 3 (ii) 7 16 (iii) 7 48 (iv) 41 48 ] Solution AB 'AB 5P( ) 8AB 7P( ') 24AB (i)     P = P P 'BA BA B   from diagram 57= 82 4 1= 3 (ii) Since A and B are independent, A and B’ are independent     P' = P P 'AB A B So    P'P= P' ABA B  7 24= 11 3 7= 16 OR      P= P + P PAB A B AB    PPP P ABA B since A and B are independent events    51 1 P P83 3 7P 16 AA A    (iii) Since A and B are independent events,   P = P PAB A B 71 16 3 7 48 (iv)    P' ' = 1 PA BA B  7 1 48 41 48 A B A B A B

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 2 of 16 2 SAJC Prelim 9740/2010/02/Q7a A & B are two events with non-zero probability. Explain if each of the following statements is necessarily true, necessarily false, or neither necessarily true nor necessarily false. (i) If A & B are mutually exclusive, then they are independent. [1] (ii) If A & B are independent, then they are mutually exclusive. [1] Solution (i) If A & B are mutually exclusive, then () 0PA B But since P( ) 0A  and P( ) 0B  , () 0 P ( ) P ( )P AB A B  i.e. A and B are not independent. Statement is necessarily false. OR P( )P( | ) 0 (since and are mutually exclusive)P( ) If and are independent, then P( | ) P( ) 0 contradiction!! ABAB A B B AB A B A   (ii) If A & B are independent, then P( ) P( )P( ) 0AB A B  since P( ) 0, P( ) 0.AB Hence, P( ) 0 . and are not mutually exclusive. AB AB AB      Statement is necessarily false.

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 3 of 16 3 DHS JC2 Prelim 9758/2019/02/Q7 In a school survey, a group of 80 students ar e asked about how much time per week (to nearest hour) they spend on their co-curricular activities (CCA). The readings are shown below: CCA (hours) 3 or less 4 to 6 7 or more Boy 17 20 10 Girl 18 15 k k A student is select ed random from the group. Defining the events as follows: G : The student is a girl. L : The student spends 6 hours or less weekly. M : The student spends 4 hours or more weekly. Find the following probabilities in terms of .k (i) P( ) 'L M [2] (ii) P( ')GL [1] (iii) Given that 2P( ) 5LM , find the value of .k Hence determine if L and M are independent, justif ying your answer. [3] (iv) If the events G and ( )LM are mutually exclusive, find the value of .k [1] [(i) 45 80 k (ii) 10 k k (iii) 3k  ; not independent (iv) 15k  ] Solution (i)  80 ( )P( ' ') 80 80 35 45 80 80 nL MLM k k    OR P( ' ') P( ') ( ') P( ' ') 10 35 080 80 45 80 LM L P M LM k k       (ii) (' )P( ') (' ) 1 0 P GL kGL PL k   (iii) Given 2P( ) 5LM 4 to 6 hours

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 4 of 16 From table: 20 (15 ) 35P( ) 80 80 kkLM    Solving, we get 3k  67 45 603 2P( )P( ) 80 80 1280 5LM   Since P( ) P( )P( )L ML M , L and M are not independent. OR 70 67P( ) 80 80 kL  and 35 32 67P( ) 45 45 80 kLM   Since P( ) P( )L LM , L and M are not independent. (iv) Since P( ( )) 0GL M  15 0k  15k 

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 5 of 16 4 9205/2003/02/Q11 Hari, a car salesman, has found that he can make a sale to 65% of his male customers but to only 45% of his female customers. All of Hari’s sales are independent. On Tuesday morning Hari has two male customers and one female customer. Find the probability that Hari makes exactly two sales. [4] 60% of Hari’s customers are male. Find the probability that, on Wednesday morning, Hari makes a sale to his first customer. [3] Find the probability that, on Thursday morning, Hari makes his first sale to his fourth customer. [3] For a randomly chosen customer, find the probab ility that the customer is female given that Hari makes a sale to that customer. [4] [ 0.437 125; 0.57 ; 0.0453; 6 19 ] Solution P (makes exactly 2 sales) P(sale to both males, not female) + P(sale to female and only one out of two males) 2(0 65) (0 55) (0 45)(0 65)(0 35)2! 0.437 125      P(sale to first customer) (0.6)(0.65) (0.4)(0.45) 0.57 P(first sale to fourth customer) 3 3 (1 0.57) (0.57) (0.43) (0.57) 0.0453 (3sf)    P(customer is female | sale to the customer) = P(customer is female and Hari makes a sale) P(Hari makes a sale) (0.4)(0.45) 6 0.57 19

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 6 of 16 5 9758/2020/02/Q8 In a game, a computer randomly chooses 12 shapes from 11 circles and 17 rectangles. The number of rectangles chosen is denoted by R. (i) Show that P( R = 1) < P(R = 2). [2] The number of rectangles available is now increased by r. The computer randomly chooses 12 shapes from the 11 circles and (17 + r) rectangles. The probability that 4 rectangles are chosen is now 15 times the probability that 3 rectangles are chosen. (ii) Find the value of r. [5] [(ii) 6] Solution (i) 17 11 11 1 17 1P( 1) 28 30421755 1789515 12 R         17 11 21 0 17 16 11 1496 88 1P( 2) P( 1) 28 2 30421755 30421755 1789515 1789515 12 RR              (ii) 17 11 17 11 48 39 1528 28 12 12 rr rr                      17 11 17 11 1548 39 (17 )(16 )(15 )(14 ) (17 )(16 )(15 )165 15 551234 123 14 20 6 rr rrrr rrr r r                          

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ___________________ Tutorial S1B: Probability Page 7 of 16 6 9740/2009/02/Q7 A company buys p % of its electronic components from supplier A and the remaining (100 – p) % from supplier B. The probability that a randomly chosen component supplied by A is faulty is 0.05. The probability that a randomly chosen component supplied by B is faulty is 0.03. (i) Given that p = 25, find the probability that a randomly chosen component is faulty. [2] (ii) For a general value of p, the probability that a randomly chosen component that is faulty was supplied by A is denoted by f (p). Show that  0.05f. 0.02