27

The event E can materialize in the following mutually exclusiveways:

(i) Person A is selected and democratization is introducedie A1 I E happens

(ii) Person B is selected and democratization is introducedieA2 I E happens

(iii) Person C is selected and democratization is introducedie A3 I E happens

Thus E = (A1 I E) U (A2 I E) U(A3 I E) , where these sets aredisjointHence by addition rule of probability we have P(E) = P(A1 I E) + P(A2 I E) + P(A3 I E)

= P(A1) P(E/A1) + P(A2) P(E/A2) + P(A3) P(E/A3)

=94

× 0.3 +92

× 0.5 +93

× 0.8

=9046 =

4523

Example 20:In a bolt factory machines A1, A2, A3 manufacture

respectively 25%, 35% and 40% of the total output. Of these 5, 4,and 2 percent are defective bolts. A bolt is drawn at random fromthe product and is found to be defective. What is the probabilitythat it was manufactured by machine A2 ?Solution:P(A1 ) = P( that the machine A1 manufacture the bolts) = 25% = 0.25Similarly P(A2) = 35% = 0.35 and P(A3) = 40% = 0.40Let B be the event that the drawn bolt is defective. P(B/ A1 ) = P (that the defective bolt from the machine A1 )

= 5 % = 0.05 Similarly, P(B/ A2 ) = 4% = 0.04 And P(B/ A3) = 2% = 0.02

28

We have to find P(A2/ B).Hence by Bayes’ theorem, we get

P(A2/ B).=)A/B(P)A(P)A/B(P)A(P)A/B(P)A(P

)A/B(P)A(P

332211

22

++

=)02.0)(4.0()04.0)(35.0()05.0)(25.0(

)04.0)(35.0(++

= 2869

= 0.4058

Example 21:A company has two plants to manufacture motorbikes.

Plant I manufactures 80 percent of motor bikes, and plant IImanufactures 20 percent. At Plant I 85 out of 100 motorbikes arerated standard quality or better.At plant II only 65 out of 100 motorbikes are rated standard qualityor better.(i) What is the probability that the motorbike, selected at

random came from plant I. if it is known that the motorbike isof standard quality?

(ii) What is the probability that the motorbike came from plant IIif it is known that the motor bike is of standard quality?

Solution:Let A1 be the event of drawing a motorbike produced by plant I.A2 be the event of drawing a motorbike produced by plant II.B be the event of drawing a standard quality motorbike producedby plant I or plant II.Then from the first information, P(A1) = 0.80, P(A2) = 0.20From the additional informationP(B/A1) = 0.85P(B/A2) = 0.65The required values are computed in the following table.The final answer is shown in last column of the table.