Question

Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.

Solution

Let E₁ be the event of choosing the bag I, E₂ the event of choosing the bag II and A be the event of drawing a red ball.
Then

straight P left parenthesis straight E subscript 1 right parenthesis space space equals space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 half

Now,

straight P left parenthesis straight A space vertical line thin space straight E subscript 1 right parenthesis space equals space straight P left parenthesis drawing space straight a space red space ball space from space Bag space straight I right parenthesis space equals space 3 over 7

and

Now, the probability of drawing a ball from Bag If, being given that it is red, is P(E₂| A).
By using Bayes’ theorem, we have

straight P left parenthesis straight E subscript 2 space vertical line space straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A space vertical line space straight E subscript 2 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A space vertical line space straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A space vertical line space straight E subscript 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space equals space fraction numerator begin display style 1 half end style cross times begin display style 5 over 11 end style over denominator begin display style 1 half end style cross times begin display style 3 over 7 end style plus begin display style 1 half end style cross times begin display style 5 over 11 end style end fraction space equals fraction numerator begin display style 5 over 22 end style over denominator begin display style 3 over 14 end style plus begin display style 5 over 22 end style end fraction space equals space fraction numerator begin display style 5 over 22 end style over denominator begin display style fraction numerator 33 plus 35 over denominator 154 end fraction end style end fraction space equals space fraction numerator begin display style 5 over 22 end style over denominator begin display style 68 over 154 end style end fraction

equals space 5 over 22 cross times 154 over 68 space equals 35 over 68

For Daily Free Study Material Join wiredfaculty

Probability

Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.

Some More Questions From Probability Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location