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Let E₁, E₂, E₃ be three independent events associated with a random experiment such that 3 P (E₁ E ₂ E ₃ )=P ( E ₁ E₂ E ₃ )=9 P ( E ₁ E ₂ E₃ )=3-3 P (E₁ E₂ E₃ ) , where P (E₁ ), P (E₂ ), P (E₃ ) 1 and P(A) denotes probability of event A . If absolute value of | array lll P (E₁ ) & P (E₂ ) & P (E₃ ) P (E₂ ) & P (E₃ ) & P (E₁ ) P (E₃ ) & P (E₁ ) & P (E₂ ) array |= a b , where a, b N , then find the least value of (a+b)

Correct answer

41

Step-by-step solution

Let P (E₁ )=a, P (E₂ )=b and P (E₃ )=c aligned & 3 a(1-b)(1-c)=(1-a) b(1-c)=9(1-a)(1-b) c=3(1-a)(1-b)(1-c) & 3 a 1-a = b 1-b = 9 c 1-c =3 a= 1 2 , b= 3 4 , c= 1 4 & Now, | array lll 1 / 2 & 3 / 4 & 1 / 4 3 / 4 & 1 / 4 & 1 / 2 1 / 4 & 1 / 2 & 3 / 4 array |= 1 64 | array lll 2 & 3 & 1 3 & 1 & 2 1 & 2 & 3 array |=- 9 32 aligned a b = 9 32 a+b=41

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