A missile target may be at a point P with probability 9 10 or at a point Q with probability 1 10 we have 20 shells each of which can be fired either at point P or Q . Each shell may hit the target independently of the other shoot with probability 2 3 . Then number of shells must be fired at point P to hit any target with maximum probability is

Step-by-step solution

Let A be the event that target is hit E₁, E₂ be the events to select target P and Q respectively aligned & Then P (E₁ )= 9 10 , P (E₂ )= 1 10 & P ( A E₁ )=1- ( 1 3 )^x, P ( A E₂ )=1- ( 1 3 )^ 20-x where x number of shells used to select target P & Then P(A)=P (E₁ ) P ( A E₁ )+P (E₂ ) P ( A E₂ )= 9 10 (1- ( 1 3 )^x )+ 1 10 (1- ( 1 3 )^ 20-x ) & d P(A) d x = 9 10 x- ( 1 3 )^x - _c 3+ 1 10 ( 1 3 )^ 20-x 1 - _ e 3=0 & _c 3 10 (9.3^ -x -3^ x-20 )=0 & 3^ 2-x =3^ x-20 x=11 aligned