Get live Maths 1-on-1 Classs - Class 6 to 12

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Chapter 13 Class 12 Probability

Serial order wise

Last updated at March 16, 2023 by Teachoo

Example 33 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)= (𝑃(𝐸3).𝑃(𝐴|𝐸3))/(𝑃(𝐸1)𝑃(𝐴|𝐸1)+𝑃(𝐸2)𝑃(𝐴|𝐸2)+𝑃(𝐸3)𝑃(𝐴|𝐸3)+𝑃(𝐸4)𝑃(𝐴|𝐸4)) P(E1) = Probability that ball drawn is from box I = 1/4 P(A|E1) = Probability of that black ball is selected from Box I = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑥) = 3/18 P(E2) = Probability that ball drawn is from box II = 1/4 P(A|E2) = Probability of that black ball is selected from Box II = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑥) = 2/8 = 1/4 P(E3) = Probability that ball drawn is from box III = 1/4 P(A|E3) = Probability of that black ball is selected froAm Box II = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑥) = 1/7 P(E4) = Probability that ball drawn is from box IV = 1/4 P(A|E4) = Probability of that black ball is selected from Box IV = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑙𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑥) = 4/13 Putting values in Equation "P(E3|A)"=(𝑃(𝐸3).𝑃(𝐴|𝐸3))/(𝑃(𝐸1)𝑃(𝐴|𝐸1)+𝑃(𝐸2)𝑃(𝐴|𝐸2)+𝑃(𝐸3)𝑃(𝐴|𝐸3)+𝑃(𝐸4)𝑃(𝐴|𝐸4)) = (1/4 × 1/7)/( 1/4 × 3/18 + 1/4 × 1/4 + 1/4 × 1/7 + 1/4 × 4/13 ) = (1/28)/( 1/24 + 1/16 + 1/28 + 1/13 ) = (1/28)/( (1/(4 × 6) + 1/(4 × 4) + 1/(4 × 7)) + 1/13) = (1/28)/( ((4 × 7 + 6 × 7 + 6 × 4)/(4 × 6 × 4 × 7)) + 1/13) = (1/28)/((28 + 42 + 24)/(4 × 6 × 4 × 7) + 1/13) = (1/28)/(94/(4 × 6 × 4 × 7) + 1/13) = (1/28)/((94 × 13 + 4 × 6 × 4 × 7)/(13 × 4 × 6 × 4 × 7) ) = 1/((94 × 13 + 4 × 6 × 4 × 7)/(13 × 4 × 6) ) = (13 × 4 × 6)/( 94 × 13 + 4 × 6 × 4 × 7) = (13 × 2 × 6)/( 47 × 13 + 2 × 6 × 4 × 7) = 156/(611 + 336) = 156/947 = 0.164