JEE Advanced 2019 · Paper 1 · Mathematics Q15

MathematicsNumerical+3Question 51 of 54

Let $S$ be the sample space of all $3\times3$ matrices with entries from the set $\{0, 1\}$ . Let the events $E_1$ and $E_2$ be given by

$E_1 = \{A\in S : \det A = 0\}$ and

$E_2 = \{A\in S : \text{sum of entries of } A \text{ is } 7\}$ .

If a matrix is chosen at random from $S$ , then the conditional probability $P(E_1|E_2)$ equals ______

Your answer

Answer and worked solution

Official answer: 0.5

Size of $E_2$ . A sum of 7 means exactly two zeros: $\binom92=36$ matrices.

Both zeros in one row. The other two rows are both $(1,1,1)$ , so $\det=0$ : $3\times3=9$ matrices.

Zeros in two different rows. The third row is $(1,1,1)$ .

If the zeros are in the same column, the two rows are equal and $\det=0$ : $3\times3=9$ matrices.
If the zeros are in different columns, e.g. $(0,1,1),(1,0,1),(1,1,1)$ , then $\det=1\ne0$ .

Probability.
$P(E_1\mid E_2)=\frac{18}{36}=0.5.$
Answer: 0.5

JEE Advanced 2019 · Paper 1, Mathematics question 15