JEE Advance - Mathematics (2019 - Paper 1 Offline - No. 13)
Let S be the sample space of all 3 $$ \times $$ 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given by
E1 = {A$$ \in $$S : det A = 0} and
E2 = {A$$ \in $$S : sum of entries of A is 7}.
If a matrix is chosen at random from S, then the conditional probability P(E1 | E2) equals ...............
E1 = {A$$ \in $$S : det A = 0} and
E2 = {A$$ \in $$S : sum of entries of A is 7}.
If a matrix is chosen at random from S, then the conditional probability P(E1 | E2) equals ...............
Answer
0.5
Explanation
Given sample space (S) of al 3 $$ \times $$ 3 matrices with entries from the set {0, 1} and events
E1 = {A$$ \in $$S : det(A) = 0} and
E2 = {A$$ \in $$S : sum of entries of A is 7}.
For event E2, means sum of entries of matrix A is 7, then we need seven 1s and two 0s.
$$ \therefore $$ Number of different possible matrices = $${{9!} \over {7!2!}}$$
$$ \Rightarrow $$ n(E2) = 36
For event E1, |A| = 0, both the zeroes must be in same row/column.
$$ \therefore $$ Number of matrices such that their determinant is zero = $$6 \times {{3!} \over {2!}}$$ = 18 = $$n({E_1} \cap {E_2})$$
$$ \therefore $$ Required probability,
$$P\left( {{{{E_1}} \over {{E_2}}}} \right) = {{n({E_1} \cap {E_2})} \over {n({E_2})}}$$
$$ = {{18} \over {36}}$$
$$ = {1 \over 2} = 0.50$$
E1 = {A$$ \in $$S : det(A) = 0} and
E2 = {A$$ \in $$S : sum of entries of A is 7}.
For event E2, means sum of entries of matrix A is 7, then we need seven 1s and two 0s.
$$ \therefore $$ Number of different possible matrices = $${{9!} \over {7!2!}}$$
$$ \Rightarrow $$ n(E2) = 36
For event E1, |A| = 0, both the zeroes must be in same row/column.
$$ \therefore $$ Number of matrices such that their determinant is zero = $$6 \times {{3!} \over {2!}}$$ = 18 = $$n({E_1} \cap {E_2})$$
$$ \therefore $$ Required probability,
$$P\left( {{{{E_1}} \over {{E_2}}}} \right) = {{n({E_1} \cap {E_2})} \over {n({E_2})}}$$
$$ = {{18} \over {36}}$$
$$ = {1 \over 2} = 0.50$$
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