Q. 295.0( 1 Vote )

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 box III.

Answer :

Given:


Box I has 3 Black, 4 White, 5 Red, 6 Blue balls


Box II has 2 Black, 2 White, 2 Red, 2 Blue balls


Box III has 1 Black, 2 White, 3 Red, 1 Blue balls


Box IV has 4 Black, 3 White, 1 Red, 5 Blue balls


Let us assume U1, U2, U3, U4 and A be the events as follows:


U1 = choosing Box I


U2 = choosing Box II


U3 = choosing Box III


U4 = choosing Box IV


A = choosing Black ball from box


We know that each urn is most likely to choose. So, probability of choosing a urn will be same for every Urn.






The Probability of choosing balls from each box differs from box to box and the probabilities are as follows:


P(A|U1) = P(Choosing black ball from Box I)





P(A|U2) = P(Choosing black ball from Box II)






P(A|U3) = P(Choosing black ball from Box III)





P(A|U3) = P(Choosing black ball from Box III)





Now we find


P(U3|A) = P(The black ball is from BallIII)


Using Baye’s theorem:







The required probability is .


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