SAJC Check your Understanding Probability teacher

Check your Understanding (Probability) Section 1: Venn Diagram 1. Kesamet Junior College celebrated Staff Fruits Day, where 150 teachers received a free fruit each. The gender of the teachers and types of fruits each teacher received were recorded in the table shown below. Male Female Apple 12 28 Banana 55 32 Orange 8 15 One of the teachers is selected at random. M is the event that the teacher selected is a male. A is the event that the teacher selected received an apple. B is the event that the teacher selected received a banana. C is the event that the teacher selected received an orange. (i) Find the following probabilities. (a) P( )MA , [2] (b) P( )MB . [2] (c) P( | )MC , [2] (ii) Determine whether the events M and C are independent, justifying your answer. [2] (ia) 1 4 2 103P( ) P( ) P( ) P( ) 2 15 25 150M A M A M A         Or 12 55 8 28 103P( ) 150 150MA     

(ib) 28 15 43P( ) 150 150MB    (ic) 8P( ) 8 150P( | ) 23P( ) 23 150 MCMC C    or n( ) 8P( | ) n( ) 23 MCMC C  (ii) 8P( | ) 23MC  but 1P( ) 2M   i.e. P( | ) P( )M C M  M and C not independent 2. In the year 2016, there were 620 male students and 520 female students in the JC1 cohort of Hwa Chong Institution (HCI). The number of these students studying H2 Mathematics, H2 Chemistry and H2 Physics are listed in the table below. One of the students is chosen at random. Events M, C and F are defined as follows: M : The student chosen is studying H2 Mathematics. C : The student chosen is studying H2 Chemistry. F : The student chosen is a female. Find (i)  'P MF , [1] (ii)  'P| MF , [2] (iii)  P CF [2] (iv) Give a reason in each case, state whether the following statements is True or False. (a) The events M and F are independent. [2] (b) The events M and C are not mutually exclusive. [1] 400 students were enrolled into HCI via the Joint Admission Exercise (JAE) in 2016. It was given that 25% of female students and 40% of male students who study H2 Mathematics were students who enrolled via the JAE. H2 Mathematics H2 Chemistry H2 Physics Male 610 480 306 Female 512 440 200 Total 1122 920 506

Three students who were enrolled via the JAE were chosen at random. (v) Find the probability all three of them are female students who study H2 Mathematics. [2] (vi) Find the probability that none of them study H2 Mathematics. [2] 2 (i) Total number of students 620 520 1140  ' 610 61P or 0.5351140 114MF   (ii)       ' ' ' P P| P MF MF F   610 610 611140 or 0.984620 620 62 1140    (iii)        P P P PC F C F C F     920 520 440 1140 1140 1140 1000 1140     50 57 or 0.877 (iv) (a) False.    ' 61 1122P | P 62 1140M F M    M and 'F are not independent and hence M and F are not independent OR    ' 1122 620 5797PP 1140 1140 10830MF     ' 61P 114MF Since      ''P P PM F M F   M and F are not independent.

(iv)( b) True.  P0MC as 1122 920 1140 Therefore, events M and C are not mutually exclusive. (v) Number of female JAE students who study Mathematics. 0.25(512) 128 Required probability = 128 127 126 0.0322400 399 398    OR Required probability 128 3 0.0322400 3    (vi) Number of JAE male students who study Mathematics (0.40)(610) 244 JAE students who do not study Mathematics 400 128 244 28    Required probability = 28 27 26 0.000309400 399 398    OR Required probability 28 3 0.000309400 3    3. In a certain junior college, each student own exactly one mobile phone: either a Xiphone or a Samsong phone. The number of male and female students who own either a Xiphone or a Samsong phone is shown in the table below. Xiphone Samsong Male 227 318 Female 263 212 A student from this junior college is chosen at random. Let F be the event that the student is a female and X be the event that the student owns a Xiphone. (i) Find

(a) P(F), [2] (b) P( )FX , [1] (c) P( ')FX . [1] (ii) Determine whether the events F and X independent, justifying your answer. [2] (iii) Given that the student selected owns a Samsong, find the probability that the student is a male. [2] A telco company selects two students from the junior college for a survey. If each student has the same chance of being selected, find the probability that one of the students selected has a Xiphone and the other has a Samsong. [3] 3(ia) No of females = 263 + 212 Total no. of people = 227 + 263 + 318 + 212 = 1020   475 0.46568 0.4661020PF    (ib)   263 0.25784 0.2581020P F X    (ic)   475 318 739' 1020 1020P F X        475 490 0.223712 0.2241020 1020P F P X     From (ii)   0.258 0.224P F X   Hence  P F X    P F P X Hence F and X are not independent       '''' ' P F XP F X PX  318 530 = 0.6 Prob (one Xiphone and one Samsong) = 490 5302 1020 1019 =0.4997=0.500 Alternative: 490 530 11 1020 1 CC C 4. For events A and B, it is given that  P 0.2AB and  P | 0.4AB  . (a) Find  P AB . [3] It is further given that  P 0.3AB . (b) Determine whether A and B are independent. [4]

4(a)  Let P 0.40.2 0.4 0.08 2 15 A B x x x xx x     (b)  P 1 0.2 0.3 0.5A       21P 0.2 15 3B          11P P 0.5 36AB     2P 15AB Since      P P P , and are not independent.A B A B A B 5. (i) A and B are two events such that  P 0.33AB and  P 0.15AB . Show that  P 0.52A  . [1] (ii) Given further that  P 0.45B  , find  P AB and  P AB  . [3] (iii) There is another event C such that  P 0.25C  and  P 0.7AC . Determine whether events A and C are mutually exclusive. [2] 5(i)      P 1 P P 1 0.33 0.15 0.52 A A B A B            (ii)      P P P 0.45 0.33 0.12 A B B A B            P P P 0.52 0.12 0.4 A B A A B     

Alternative      P 1 P P 1 0.45 0.15 0.4 A B B A B                P P P 0.52 0.4 0.12 A B A A B       (iii)        P P P P 0.52 0.25 0.7 0.07 A C A C A C         Since  P 0.07 0AC   , A and C are not mutually exclusive events. 6. For events A and B, it is given that P( ) 0.7,A  P( ) 0.5B  and P( ) 0.8.AB  (i) State, with a reason, whether A and B are mutually exclusive. [2] For a third event C, it is given that P( ' ' ) .A B C q   (ii) When 0.1,q find P( ' ' ').A B C [2] (iii) State the maximum value of q. [1] 6(i) P( ) P( ) P( ) 0.4 0A B A B B     Hence A and B are not mutually exclusive. 6(ii) P( ) P( ) P( ) P( ) 0.7 0.5 0.4 0.8 AB A B A B          P( ' ' ') 1 P( ) 1 [P( ) P( ' ' )] 1 [0.8 0.1] 0.1 A B C A B C A B A B C                6(iii) Max q = 0.2

Section 2: Tree Diagram 1. The probability that a laboratory mouse has a particular disease is p. A test