Question 3 - Case Based Questions (MCQ) - Chapter 13 Class 12 Probability

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Question In answering a question on a multiple choice test for class XII, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assume that a student who guesses at the answer will be correct with probability 1/3. Let E1, E2, E be the events that the student knows the answer, guesses the answer and answers correctly respectively. Based on the above information, answer the following:Given E1 : Student knows the answer E2 : Student guesses the answer E : Student answers correctly Now, P("E1") = P(student knows the answer) = 𝟑/𝟓 P(E2) = P(student guesses the answer) = 𝟐/𝟓 Also, Given that assume that a student who guesses at the answer will be correct with probability 1/3 i.e. P(answer correct | he guesses) ∴ P(E | E2) = 1/3 Question 1 What is the value of P(E1)? (a) 2/5 (b) 1/3 (c) 1 (d) 3/5P(E1) = 𝟑/𝟓 So, the correct answer is (d) Question 2 Value of P(E | E1) is (a) 1/3 (b) 1 (c) 2/3 (d) 4/5 P(E | E1) = P( student answer correct | he knows answer) = Probability that student answers correctly, if he knows the answer = 1 So, the correct answer is (b) Question 3 ∑_(𝑘=1)^(𝑘=2)▒〖𝑃(𝐸│𝐸_𝑘 ) 𝑃(𝐸_𝑘 ) 〗 Equal (a) 11/15 (b) 4/15 (c) 1/5 (d) 1 ∑_(𝑘=1)^(𝑘=2)▒〖𝑃(𝐸│𝐸_𝑘 ) 𝑃(𝐸_𝑘 ) 〗 = 𝑃(𝐸│𝐸_1 ) 𝑃(𝐸_1 )+ 𝑃(𝐸│𝐸_2 ) 𝑃(𝐸_2 ) = 𝟏×𝟑/𝟓+𝟏/𝟑×𝟐/𝟓 = 3/5×2/15 = 𝟏𝟏/𝟏𝟓 So, the correct answer is (a) Question 4 Value of ∑_(𝑘=1)^(𝑘=2)▒〖 𝑃(𝐸_𝑘 ) 〗 (a) 1/3 (b) 1/5 (c) 1 (d) 3/5 ∑_(𝑘=1)^(𝑘=2)▒𝑃(𝐸_𝑘 ) = 𝑃(𝐸_1 ) + 𝑃(𝐸_2 ) = 3/5+2/5 = 1 So, the correct answer is (c) Question 5 What is the probability that the student knows the answer given that he answered it correctly? (a) 2/11 (b) 5/3 (c) 9/11 (d) 13/3 We need to find the Probability that the student knows the answer given that he answered it correctly i.e. P(𝐸_1 "|E") Now, "P(" 𝑬_𝟏 "|E) = " (𝑃(𝐸_1 ). 𝑃(𝐸|𝐸_1))/(𝑃(𝐸_1 ). 𝑃(𝐸|𝐸_1 ) + 𝑃(𝐸_2 ). 𝑃(𝐸|𝐸_2 ) ) So, "P(" 𝑬_𝟏 "|E) = " (𝑃(𝐸_1 ). 𝑃(𝐸|𝐸_1))/(𝑃(𝐸_1 ). 𝑃(𝐸|𝐸_1 ) + 𝑃(𝐸_2 ). 𝑃(𝐸|𝐸_2 ) ) = (3/5 × 1)/(3/5 × 1 + 2/5 × 1/3 ) = 3/5×15/11 = 𝟗/𝟏𝟏 So, the correct answer is (c)