Probability Qns and Solns (SAJC)

Chapter 2: Probability 1. CJC JC2 Prelim 8865/2019/Q8 (a) Two mutually exclusive events A and B are such that ( )P A and ( )P B are both non -zero. Determine if A and B are independent events. Justify your answer. [1] (b) It is given that events C and D are such that ( )P | 0.4CD = , ( )P 0.15CD= and ( )P 0.82CD= . Find the value of ( )P C . [4] Answer: (b) 0.595 1. CJC JC2 Prelim 8865/2019/Q8 (Solutions) (a) A and B are mutually exclusive events, ( )P0AB= . However, ( )P A and ( )P B are both non-zero so ( ) ( )P P 0AB  . Since ( ) ( ) ( )P P PA B A B , A and B are not independent. (b) ( ) ( ) ( ) ( ) ( ) P | 0.4 P 0.4P 0.15 0.4P P 0.375 CD CD D D D =  = = = ( ) ( ) ( ) ( ) ( ) ( ) P 0.82 P P P 0.82 P 0.375 0.15 0.82 P 0.595 CD C D C D C C = + −  = + − = = 2. RVHS JC2 Prelim 8865/2019/Q7 Events A and B are such that ( ) ( )P P |A A B . (i) Determine, with a reason, whether A and B are independent. [1] (ii) Determine, with a reason, whether A and B are mutually exclusive. [1] (iii) Given ( ) 2P 5A = , ( ) 2P| 3AB = and ( ) 3P 8B = , and using a Venn Diagram or otherwise, find ( )P' AB . [4] Answer: (iii) 7 8 2. RVHS JC2 Prelim 8865/2019/Q7 (Solutions) (i) If A and B are independent,

( ) ( ) ( ) ( ) ( ) ( ) ( ) P P PP | P PP A B A BA B A BB = = = Since ( ) ( )P P |A A B , A, B are not independent. (ii) If A and B are mutually exclusive, ( ) ( ) ( ) ( ) P 0P | 0 PP ABAB BB = = = Thus ( ) ( )P P | 0A A B= which is not possible, therefore A, B are not mutually exclusive. Alternative Method: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) PP P | P 0 P P P 0P ABA A B B A B A B AB =       Thus not mutually exclusive. (iii) ( ) ( ) ( ) ( ) PP| P 2 3 1P 3 8 4 ABAB B AB =  =  = ( ) ( ) ( ) ( ) ( ) ( ) 2 1 3P P P ' (*) 5 4 20 3 1 1P P P ' (**) 8 4 8 A A B A B B A B A B −  =  = − = −  =  = − = ( ) ( ) 3 1 1 19P ' ' 1 P 1 (***) 20 4 8 40A B A B  = −  = − + + =  ( ) 3 1 19 7P' 20 4 40 8AB = + + = Alternative A B A B

( ) ( ) ( ) ( ) PP| P 2 3 1P (*) 3 8 4 ABAB B AB =  =  = ( ) ( ) ( ) 3 1 1P P P ' 8 4 8B A B A B−  =  = − = (**) ( ) ( ) 17P ' 1 P ' 1 88A B A B = −  = − = 3. ASRJC JC2 Prelim 8865/2019/Q7 A group of students are surveyed on whether they use any social media platforms. The numbers of students using different combinations of these platforms are shown in the above Venn diagram. The number of students who use Facebook and Twitter only is x. One of the students is chosen at random. • F is the event that the student uses Facebook. • G is the event that the student uses Instagram. • T is the event that the student uses Twitter. (i) Write down the expression for ( )P F in terms of x. [