Ex 13.1, 13 - Chapter 13 Class 12 Probability (Term 2)

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Ex 13.1, 13 An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question? Let Easy True/False questions be denoted by A1, Difficult True/False questions be denoted by B1, Easy MCQ’s be denoted by A2, & Difficult MCQ’s be denoted by B2, Given, A1 = 300 , B1 = 200 , A2 = 500 , B2 = 400 A question is selected at random We need to find the probability that it will be an easy question, given that it is a MCQ. i.e. P((A1 + A2) | (A2 + B2)) Now, P(A1 + A2) = (𝐸𝑎𝑠𝑦 𝑇𝑟𝑢𝑒 𝐹𝑎𝑙𝑠𝑒 + 𝐸𝑎𝑠𝑦 𝑀𝐶𝑄 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠) = (300 + 500)/(300 + 200 + 500 + 400) = 800/1400 = 8/14 = 4/7 P(A2 + B2) = (𝐸𝑎𝑠𝑦 𝑀𝐶𝑄 + 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡 𝑀𝐶𝑄 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠) = (500 + 400 )/(300 + 200 + 500 + 400) = 900/1400 = 9/14 Also, (A2 + B2) ∩ (A2 + B2) = A2 So, P[(A2 + B2) ∩ (A2 + B2)] = P(A2 ) = 500/1400 = 5/14 Now, P((A1 + A2) | (A2 + B2)) = 𝑃[(𝐴_1 + 𝐴_2 ) ∩ (𝐴_2 + 𝐵_2 )]/𝑃(𝐴_2 + 𝐵_2 ) = (5/14)/(9/14) = 5/14 × 14/9 = 5/9 ∴ Required probability is 𝟓/𝟗