Let S be the sample space of all 3 × 3 matrices with entries from the set $\{0, 1\}$. Let the events $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ be given by $$\begin{aligned} & E_{1}=\{A \in S: \operatorname{det} A=0\} \text { and } \\ & E_{2}=\{A \in S: \text { sum of entries of } A \text { is } 7\} . \end{aligned}$$ If a matrix is chosen at random from S, then the conditional probability $\mathrm{P}\left(\mathrm{E}_{1} \mid \mathrm{E}_{2}\right)$ equals ____

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Probability XII

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 \in S : det A = 0} and E_{2} = {A \in 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 ____

Answer: 0.5

For

E_{2}

: sum of entries is 7, meaning exactly 7 ones and 2 zeros.

n (E_{2}) =^{9} C_{2} = 36

For

E_{1} \cap E_{2}

det A = 0

when two zeros are in the same row or same column.

Number of ways to place 2 zeros in the same row:

3 \times^{3} C_{2} = 9

Number of ways to place 2 zeros in the same column:

3 \times^{3} C_{2} = 9

Total:

n (E_{1} \cap E_{2}) = 18

Therefore:

P (\frac{E _{1}}{E _{2}}) = \frac{n ( E _{1} \cap E _{2} )}{n ( E _{2} )} = \frac{18}{36} = \frac{1}{2} = 0.50

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Numerical answer