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प्रश्न
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
उत्तर
Let the events of choosing the first and second bag be E1 and E2 respectively, then
P(E1) = P(E2) = `1/2`
Let E be the event of drawing red balls, then
P(E|E1) = `4/8 = 1/2`, P(E|E2) = `2/8 = 1/4`
Now, what is the probability of drawing a ball from the first bag when it is known that it is red in colour.
= P(E1|E)
By Bayes' theorem P(E1|E) = `(P(E_1) xx P(E|E_1))/(P(E_1) xx P(E|E_1) + P(E_2) xx P(E|E_2))`
= `(1/2 xx 1/2)/(1/2 xx 1/2 + 1/2 xx 1/4)`
= `(1/4)/(1/4 + 1/8)`
= `1/4 xx 8/3`
= `2/3`
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