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. [1] (ii) Given that ( ) 32P 225FT= , show that 21x= . [1] (iii) Determine whether F and T are independent. [1] (iv) Find ( )P' FT [1] (v) Explain, in the context of this question, what is meant by ( )( )P| G T F , and find its value. [3] Two students from the group are chosen at random, without replacement. 12 67 5 Twitter Instagram Facebook 87 x 18 11 4

(vi) Find the probability that both of these students each use exactly two of the three social media platforms. [2] Answer: (i) 110 204 x x + + (iv) 14 25 (v) 119 131 (vi) 5 16 3. ASRJC JC2 Prelim 8865/2019/Q7 (Solutions) (i) Total number of students 12 87 67 11 18 5 4 204 x x = + + + + + + + =+ ( ) 12 87 11 110P 204 204 xxF xx + + + +== ++ ( ) 11P 204 xFT x += + ( ) 11 32P 204 225 xFT x + = = + 225(11 ) 32(204 ) 225 32 32(204) 225(11) 4053 21193 xx xx x + = + − = − == (iii) ( ) 12 87 11 21 131P 204 21 225F + + +== + ( ) 11 18 5 21 11P 204 21 45T + + +== + ( ) ( ) 131 11 1441 32PP 225 45 10125 225FT =  =  Since ( ) ( )( ) P PP F T F T   , F and T are not independent. (iv) ( ) 5 18 67 11 21 4 126 14P' 204 21 225 25FT + + + + + = = = + (v) ( )( )P| G T F refers to the probability that a student who uses Facebook, also uses at least one of either Instagram or Twitter [Or The probability of a student using either Instagram or Twitter or both, when he already uses Facebook.]

( )( ) ( )( ) ( ) P| 21 11 87 P 225 131P 225 119 131 G T F G T F F  ++ == = (vi) Required Probability 21 87 18 21 87 18 1 225 225 1 126 125 225 224 5 16 + + + + −= − = = 4. HCI JC2 Prelim 8865/2019/Q12 During a flu season, 120 patients who display flu symptoms consulted a doctor. Of the 120 patients, 75 are females of whom 15 have actually contracted the flu. The number of male patients who have contracted the flu is x. One of the patients is chosen at random. • A is the event that the patient is a male. • B is the event that the patient has contracted flu. (i) Write down expressions for P(B) and ( )P' AB in terms of x. [2] (ii) Explain, in the context of this question, what is meant by ( )P AB . [1] Given further that ( ) 4P 7AB = , show that 20x= . [2] (iii) Are events A and B independent? Justify your answer. [1] (iv) Two patients are chosen at random, without replacement. Find the probability that exactly one of them has flu. [2] (v) A diagnostic test for the flu is used. The test has a probability of 0.93 of giving a positive result when the patient has the flu, and a probability of 0.06 of giving a positive result when the patient does not have the flu. The test is administered to a patient chosen at random from the 120 patients. (a) Draw a tree diagram to represent this information. [2] (b) Find the probability that the patient is tested positive. [2] (c) Find the probability that the patient has contracted the flu, given that he or she is tested positive. [2]

Answer: (i) 45 120 x− (iv) 5 12 (vb) 251 800 (vc) 217 251 4. HCI JC2 Prelim 8865/2019/Q12 (Solutions) (i) ( ) 15P 120 xB += ( ) 45P' 120 xAB −= (ii) It is the probability of selecting a male patient, given that the patient has contracted flu. ( ) ( ) P 4 P7 4 15 7 7 60 4 3 60 AB B x x xx x  = =+ =+ = 20x= (Shown) (iii) For A and B to be independent, ( ) ( )PPA B A= But ( ) 45 3 4P 120 8 7A = =  . Therefore A and B are not independent events. (iv) Required probability 35 85 2120 119 5 or 0.41712 =   = A B x 45 – x 15 60

(va) (vb) Required probability 35 850.93 0.06120 120 251 or 0.31375 (exact)800 =  +  = (vc) ( ) ( ) ( ) P has flu Positive P has flu Positive P Positive 35 0.93120 251 800 217 or 0.865251 =  = = 5. ACJC JC2 Prelim 8865/2019/Q6 Balls are drawn one at a time, without replacement, from a bag containing 10 red balls, 10 blue balls and 10 yellow balls. Balls of each colour are numbered 1 to 10 and points are scored according to the numbers on the respective balls. Find the probabilities that (i) the first ball drawn scores at least 9 points, [1] (ii) the first two balls drawn each score at least 9 points, [2] (iii) the first two balls drawn score at least 18 points altogether, [2] (iv) the first two balls drawn each score at least 9 points given that they score at least 18 points altogether. [2] Answer: (i