Examples

Chapter 13 Class 12 Probability
Serial order wise

### Transcript

Example 22 Coloured balls are distributed in four boxes as shown in the following table: A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the boxs III?Let A : Event that a black ball is selected E1 : Event that the ball is selected from box I E2 : Event that the ball is selected from box II E3 : Event that the ball is selected from box III E4 : Event that the ball is selected from box IV We need to find out the Probability of the ball drawn is from box III if it is black. i.e. P(E3|A) P(E3|A)= (๐(๐ธ3).๐(๐ด|๐ธ3))/(๐(๐ธ1)๐(๐ด|๐ธ1)+๐(๐ธ2)๐(๐ด|๐ธ2)+๐(๐ธ3)๐(๐ด|๐ธ3)+๐(๐ธ4)๐(๐ด|๐ธ4)) P(E1) = Probability that ball drawn is from box I = ๐/๐ P(A|E1) = Probability of that black ball is selected from Box I = (๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐ )/(๐๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐  ๐๐ ๐กโ๐ ๐๐๐ฅ) = ๐/๐๐ P(E2) = Probability that ball drawn is from box II = ๐/๐ P(A|E2) = Probability of that black ball is selected from Box II = (๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐ )/(๐๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐  ๐๐ ๐กโ๐ ๐๐๐ฅ) = 2/8 = 1/4 P(E3) = Probability that ball drawn is from box III = ๐/๐ P(A|E3) = Probability of that black ball is selected from Box II = (๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐ )/(๐๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐  ๐๐ ๐กโ๐ ๐๐๐ฅ) = ๐/๐ P(E4) = Probability that ball drawn is from box IV = ๐/๐ P(A|E4) = Probability of that black ball is selected from Box IV = (๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐ )/(๐๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐  ๐๐ ๐กโ๐ ๐๐๐ฅ) = ๐/๐๐ Putting values in Equation "P(E3|A)"=(๐(๐ธ3).๐(๐ด|๐ธ3))/(๐(๐ธ1)๐(๐ด|๐ธ1)+๐(๐ธ2)๐(๐ด|๐ธ2)+๐(๐ธ3)๐(๐ด|๐ธ3)+๐(๐ธ4)๐(๐ด|๐ธ4)) = (๐/๐ ร ๐/๐)/( ๐/๐ ร ๐/๐๐ + ๐/๐ ร ๐/๐ + ๐/๐ ร ๐/๐ + ๐/๐ ร ๐/๐๐ ) = (1/28)/( 1/24 + 1/16 + 1/28 + 1/13 ) = (1/28)/( (1/(4 ร 6) + 1/(4 ร 4) + 1/(4 ร 7)) + 1/13) = (1/28)/( ((4 ร 7 + 6 ร 7 + 6 ร 4)/(4 ร 6 ร 4 ร 7)) + 1/13) = (1/28)/((28 + 42 + 24)/(4 ร 6 ร 4 ร 7) + 1/13) = (1/28)/(94/(4 ร 6 ร 4 ร 7) + 1/13) = (1/28)/((94 ร 13 + 4 ร 6 ร 4 ร 7)/(13 ร 4 ร 6 ร 4 ร 7) ) = 1/((94 ร 13 + 4 ร 6 ร 4 ร 7)/(13 ร 4 ร 6) ) = (13 ร 4 ร 6)/( 94 ร 13 + 4 ร 6 ร 4 ร 7) = (13 ร 2 ร 6)/( 47 ร 13 + 2 ร 6 ร 4 ร 7) = 156/(611 + 336) = 156/947 = 0.164