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A bag contains a total of 20 books on physics and mathematics, Anypossible combination of books is equally likely. Ten books are chosen from the bag and it is found that it contains 6 books of mathematics. Find out the probability that the remaining books in the bag contains 3 books on mathematics. |
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Answer» Solution :Let `E_(i)(i=0,1,2,...,20)` be the event that the bag contains I books on MATHEMATICS. Since all these events are equally likely and mutually EXCLUSIVE and exhaustive, so `P(E_(1))=1//21(i=0,1,2,....,20)`and let A be the event that a drw of 10 books contains 6 books on matematics Then So, USING Bay's THEOREM, `P(A)=underset(i=0)overset(20)sumP(E_(i)).P(A//E_(i))` `=1/21[underset(i=0)overset(20)sumP(A//E_(i))]` `=1/21[underset(i=6)overset(16)sum(""^(i)C_(6)xx""^(20-i)C_(4))/(""^(20)C_(10))]` Now, we want that the bag should contain 2 more books on mathematics, i.e., `E_(8)` must occur. Using Baye's theorem, `P(E_(8)//A)=(P(E_(8))P(A//E_(8)))/(P(A))` `=((""^(8)C_(6)xx""^(12)C_(4))/(""^(20)C_(10)))/(underset(i=6)overset(16)sum((""^(i)C_(6)xx""^(20-i)C_(4))/(""^(20)C_(10))))` `=(""^(8)C_(6)xx""^(12)C_(4))/(underset(i=6)overset(16)sum(""^(i)C_(6)xx""^(20-i)C_(4)))` |
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