A random variable (X ) has the probability distribution as given below. Let (E=[X X ) is prime number] and (F= X X < 4 ), then (P(E F)= ) ( array |c|c|c|c|c|c|c|c|c| X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 P(X) & K & 2 K & K^2 & 2 K^2 & 5 K^2 & K & K & 2 K array )

Step-by-step solution

Given probability distribution is ( array c|c|c|c|c|c|c|c|c X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 P(X) & K & 2 K & K^2 & 2 K^2 & 5 K^2 & K & K & 2 K array ) ( aligned & P(X) & =1 & 8 K^2+7 K & =1 & 8 K^2+7 K-1 & =0 & 8 K^2+8 K-K-1 & =0 & 8 K(K+1)-1(K+1) & =0 & K= 1 8 as K & > 0 . aligned ) ( ) The events (given) (E= x x ) is a prime (= 2,3,5,7 ) and (F= X X < 4 = 1,2,3 ) ( aligned & P(E F)=P(E)+P(F)-P(E F) & = (2 K+K^2+5 K^2+K )+ (K+2 K+K^2 )- (2 K+K^2 ) & =6 K^2+4 K=6 1 64 + 4 8 = 38 64 aligned ) Hence, option (a) is