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Example 16 Bag I contains 3 red & 4 black balls while another Bag II contains 5 red & 6 black balls. One ball is drawn at random from one of bags & it is found to be red. Find the probability that it was drawn from Bag II. Let E1 : Bag selected is bag 1 E2 : Bag selected is bag 2 A : Ball selected is Red B : Ball selected is Black We need to find P(ball was drawn from bag 2, if ball is red) = P(E2|A) We need to find, 𝑷(𝑬𝟐"|" 𝑨)=(𝑷(𝑨) 𝑷(𝑨"|" 𝑬𝟐))/(𝑷(𝑬𝟏) 𝑷(𝑨"|" 𝑬𝟏) + 𝑷(𝑬𝟐) 𝑷(𝑨"|" 𝑬𝟐)) "P(E1)" = Probability bag selected is Bag I = 𝟏/𝟐 "P(E2)" = Probability bag selected is Bag II = 𝟏/𝟐 "P(A|E2)" = Probability red ball was selected from Bag II = 5/(5 + 6) = 𝟓/𝟏𝟏 "P(A|E1)" = Probability red ball was selected from Bag I = 3/(3 + 4) = 𝟑/𝟕 Putting values in formula, P("E2|"A) = (𝟏/𝟐 × 𝟓/𝟏𝟏)/( 𝟏/𝟐 × 𝟑/𝟕 + 𝟏/𝟐 × 𝟓/𝟏𝟏 ) = (1/2 × 5/11)/(1/2 [ 3/7 × 5/11 ]) = (5/11)/((33 + 35)/77) = (5/11)/( 68/77 ) = 5/11 × 77/68 = 𝟑𝟓/𝟔𝟖 Therefore, required probability is 𝟑𝟓/𝟔𝟖 = (1/2 × 5/11)/(1/2 [ 3/7 × 5/11 ]) = (5/11)/((33 + 35)/77) = (5/11)/( 68/77 ) = 5/11 × 77/68 = 𝟑𝟓/𝟔𝟖 Therefore, required probability is 𝟑𝟓/𝟔𝟖

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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo