Let (a, b) be the outcome of throwing a pair of fair dice. If the probability for which _ x 0 (( x)^a ) x^b exist and is finite can be expressed as p q , where p and q are co-prime positive integers, then find the value of (p+q) .

Step-by-step solution

_ x 0 (( x)^a ) x^b = _ x 0 a ( x) x^b Now applying L^ Hospital rule _ x 0 -a x b x^ b-1 = _ x 0 x x -a b 1 x^ b-2 = _ x 0 -a b 1 x^ b-2 Now for limit to be finite aligned b-2 & = 0,-1,-2,-3,-4, . b & = 2,1,0,-1,-2,-3, . . aligned But b can only be b= 2,1] as it is an outcome of a dice. Now probability is aligned & P= No. of ways to select ' a ' Total no. of ways to select ' a ' No. of ways to select ' b ' Total no. of ways to select ' b ' & P= 6 6 2 6 = 1 3 p+q=1+3=4 aligned