Suppose that a bag (A ) contains (n ) red and 2 black balls and another bag (B ) contains 2 red and (n ) black balls. One of the two bags is selected at random and two balls are drawn from it at a time. When it is known that the two balls drawn are red, if the probability that those two balls drawn are from bag (A ) is ( 6 7 ), then (n= )

Step-by-step solution

Let (E₁ ) be the event that the ball is drawn from bag (A. E₂ ) be the event that it is drawn from bag ((B) ) and (E ) that ball is red. Given, (P ( E₁ E )= 6 7 ) By Baye's, theorem, ( gathered P ( E₁ A ) 1 2 ( ^n C₂ ^ n+2 C₂ ) 1 2 ^n C₂ ^ n+2 C₂ + 1 2 ^2 C₂ ^ n+2 C₂ = 6 7 1 2 ( ^n C₂ ^ n+2 C₂ ) 1 2 [ ^n C₂ ^ n+2 C₂ + ^2 C₂ n+2 ] = 6 7 n(n-1) 2 2+n(n-1) 2 = 6 7 gathered ) ( aligned & n(n-1) n^2-n+2 = 6 7 & 7 (n^2-n )=6 n^2-6 n+12 & 7 n^2-7 n-6 n^2+6 n=12 & n^2-n=12 & n^2-n-12=0 & n^2-4 n+3 n-12=0 & n(n-4)+3(n-4)=0