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Chapter 13 Class 12 Probability
Serial order wise

Question

A factory has 3 machines X, Y and Z, producing 1000, 2000 and 3000 bolts per day respectively. The machine X  produces 1% defective bolts, Y produces 1.5% defective bolts and Z produces 2% defective bolts. At the end of the  day, a bolt is drawn at random and it is found to be defective. 

Let,

  E1 = event of drawing a bolt produced by machine X. 

  E2 = event of drawing a bolt produced by machine Y. 

  E3 = event of drawing a bolt produced by machine Z. 

  E = event of drawing a defective bolt. 

Based on the above information answer the following questions:

 

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Question 1

What is the value of P(E2)?

(A) 1/6   

(B) 1/3

(C) 1/2     

(D) 1/4

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Question 2

Find the value of P(E|E1)?

(A) 1/100 

(B) 3/200

(C) 1/50   

(D) 1/10

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Question 3

Find the value of P(E | E2)?

(A) 1/100 

(B) 3/200

(C) 1/50   

(D) 1/10

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Question 4

Find the value of P(E|E3)?

(A) 1/100 

(B) 3/200

(C) 1/50   

(D) 1/10

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Question 5

What is the probability that the drawn bolt has been produced by the machine X?

(A) 1/10 

(B) 3/200

(C) 1/20   

(D) 1/30

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Transcript

Question A factory has 3 machines X, Y and Z, producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% defective bolts and Z produces 2% defective bolts. At the end of the day, a bolt is drawn at random and it is found to be defective. Let, E1 = event of drawing a bolt produced by machine X. E2 = event of drawing a bolt produced by machine Y. E3 = event of drawing a bolt produced by machine Z. E = event of drawing a defective bolt. Based on the above information answer the following questions:Question 1 What is the value of P(E2)? (A) 1/6 (B) 1/3 (C) 1/2 (D) 1/4 Now, P(E2) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑜𝑙𝑡𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑌)/(𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑜𝑙𝑡𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑙𝑙 3 𝑚𝑎𝑐ℎ𝑖𝑛𝑒𝑠) Number of bolts produced by machine Y = 2000 Total number of defective bolts = 1000 + 2000 + 3000 = 6000 Thus, P(E2) = 2000/6000= 𝟏/𝟑 So, the correct answer is (B) Question 2 Find the value of P(E|E1)? (A) 1/100 (B) 3/200 (C) 1/50 (D) 1/10 P(E | E1) = P(defective bolt | produced by machine X) = P( defective bolt is produced by machine X) Since Machine X produces 1% defective bolts = 1% = 𝟏/𝟏𝟎𝟎 So, the correct answer is (a) Question 3 Find the value of P(E | E2)? (A) 1/100 (B) 3/200 (C) 1/50 (D) 1/10 P(E | E1) = P(defective bolt | produced by machine Y) = P( defective bolt is produced by machine Y) Since Machine Y produces 1.5% defective bolts = 1.5 % = 1.5/100 = 𝟑/𝟐𝟎𝟎 So, the correct answer is (B) Question 4 Find the value of P(E|E3)? (A) 1/100 (B) 3/200 (C) 1/50 (D) 1/10 P(E | E3) = P(defective bolt | produced by machine Z) = P( defective bolt is produced by machine Z) Since Machine Z produces 2% defective bolts = 2 % = 𝟐/𝟏𝟎𝟎 = 𝟏/𝟓𝟎 So, the correct answer is (c) Question 5 What is the probability that the drawn bolt has been produced by the machine X? (A) 1/10 (B) 3/200 (C) 1/20 (D) 1/30 We need to find Probability that the drawn defective bolt is produced by machine X i.e. P(𝑬_𝟏 "|E") 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 )+𝑃(𝐸_3 ). 𝑃(𝐸┤| 𝐸_3) ) "P(" 𝑬_𝟏 ")" = Probability that the bolt is made by machine X = 1000/6000 = 𝟏/𝟔 𝑷("E|" 𝑬_𝟏) This is calculated in Question 2 𝑃("E|" 𝐸_1) = 𝟏/𝟏𝟎𝟎 "P(" 𝑬_𝟐 ")" = Probability that the bolt is made by machine Y = 2000/6000 = 𝟏/𝟑 𝑷("E|" 𝑬_𝟐) This is calculated in Question 3 𝑃("E|" 𝐸_2) = 𝟑/𝟐𝟎𝟎 "P(" 𝑬_𝟑 ")" = Probability that the bolt is made by machine Z = 3000/6000 = 𝟏/𝟐 𝑷("E|" 𝑬_𝟑) This is calculated in Question 4 𝑃("E|" 𝐸_3) = 𝟏/𝟓𝟎 Putting values in formula, "P(" 𝐸_1 "|E) = " (1/6 × 1/100)/(1/6 × 1/100 + 1/3 × 3/200 + 1/2 × 1/50) = ( 1/600)/(1/600 + 1/200 + 1/100) = ( 1/600)/((1 + 3 + 6)/600 ) = ( 1/600)/(10/600 ) Putting values in formula, "P(" 𝐸_1 "|E) = " (1/6 × 1/100)/(1/6 × 1/100 + 1/3 × 3/200 + 1/2 × 1/50) = ( 1/600)/(1/600 + 1/200 + 1/100) = ( 1/600)/((1 + 3 + 6)/600 ) = ( 1/600)/(10/600 ) = 1/600×600/10 = 𝟏/𝟏𝟎 So, the correct answer is (a)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.