Let S be the sample space of all 3 × 3 matrices with entries from the set 0 ,   1 . Let the events E 1 and E 2 be given by E 1 = A ∈ S :   d e t   A = 0 and E 2 = A ∈ S : sum of entries of A is 7 . If a matrix is chosen at random from S , then the conditional probability P E 1 E 2 equals____
A. A
For sum to be seven, we need seven 1 s and two 0 s Number of different possible matrices= = 9 7   2 = 36 Eg. 1 1 1 1 1 1 0 0 1 arrangement of seven 1 s and two 0 s Now, for determined of matrix to be zero, both the zero must be in same row or in same column. Eg. 1 0 0 1 1 1 1 1 1 Now, arrangement of 0s and 1 s = 6 S e l e c t i o n   o f a   c o l u m n / r o w   ×    3 a r r a n g e m e n t o f   t w o   0 s a n d   o n e   1   i n   a   r o w / c o l u m n    =    18 Hence, required probability = 18 36 = 1 2 (Note: If two zero's lies on different row/column then 0 1 1 1 0 1 1 1 1 = + 1 ≠ 0 )
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