Question

2. (a) Let (a, b, c) denote the result of throwing three dice of colours, amber, blue and crimson, respectively., e.g., (1, 5, 3) represents throwing amber dice =1, blue dice = 5, crimson dice = 3. What is the probability of throwing these three dice such that the (a, b, c) satisfy the equation b² − 4ac ≥ 0? [7 marks]

(b) From a survey to assess the attitude of students in their study, 80% of them are highly motivated, 90% are hard working, and only 5% are neither highly motivated nor hard working. i. Calculate the probability that a randomly selected student is both highly motivated and hard working. [3 marks] ii. Given that the student is hard working, what is the probability that this student is not highly motivated? [3 marks]

(c) It is known from past experience that for students who studied AMA1501 for at least 100 hours, their chance of passing is 0.95, while for those who studied AMA1501 for less than 100 hours, their chance of passing is 0.1. It is also known that the probability of passing is 90%. If a student is randomly chosen and is found to have passed , what is the probability that he/she has studied for less than 100 hours? [7 marks]

Answer 1

b)

probability that a randomly selected student is  neither highly motivated nor hard workin = 0.05

probability that a randomly selected student is  either highly motivated or hard workin = 1-0.05 = 0.95

i) probability that a randomly selected student is both highly motivated and hard working =0.80+0.90-0.95= 0.75

ii)

Given that the student is hard working, probability that this student is not highly motivated = (0.90-0.75)/0.90 = 0.1667

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c)

P(   pass   | at least 100 hours)=   0.95
P(   pass   | less than 100)=   0.1

P(pass) = 0.90

let P(at least 100 hours) = x

P(less than 100) = 1-x

so, P(pass) = 0.95x + (1-x)*0.10 = 0.90

x = 0.9412

(1-x) = 1-0.9412 = 0.0588

P(less than 100 | pass) = P(less than 100)*P(pass| less than 100)/P(pass)= 0.0588*0.1/0.90=      0.0065 (answer)