Let X denote the number of elements in set X . Let S = 1 ,   2 ,   3 ,   4 ,   5 ,   6 be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S , then the number of ordered pairs A ,   B such that 1 ≤ B < A , equals
A. A
Since A and B are independent events P A ∩ B = P A P B A ∩ B S = A S · B S S   A ∩ B = A · B ∴ x = n u m b e r   o f   e l e m e n t   i n   s e t S = 6 6 A ∩ B = A B 6 n A ∩ B = n A n B s e t   x = n x Case I : n A ∩ B = 1 ⇒ n A n B = 6 a   n A = 6 ,   n B = 1 N u m b e r   o f   w a y s = C 6   6 × C 1   6 = 6 b   n A = 3 ,   n B = 1     Number   of   ways = C 3 6 × C 1 3 × C 2 5 = 180 Case I I : n A ∩ B = 2 ⇒ n A n B = 12 a   n A = 6 n B = 2 N u m b e r   o f   w a y s = C 6   6 C 2   6 = 15 b   n A = 4 n B = 3 N u m b e r   o f   w a y s = C 4   6 × C 2   4 × C 1   2 = 180 Case I I I : n A ∩ B = 3 ⇒ n A n B = 18 a   n A = 6 , n B = 3   Number   of   ways ⇒ C 6   6 × C 3   6 = 20 Case I V : n A ∩ B = 4 ⇒ n A n B = 24 a   n A = 6 n B = 4   Number   of   ways ⇒ C 6   6 × C 4   6 = 15 Case V : n A ∩ B = 5 ⇒ 1 n A n B = 30 a   n A = 6 n B = 5   Number   of   ways ⇒ C 6   6 C 5   6 = 6 Total ordered pairs = 422
Related: Mathematics — Probability · All PYQ Banks