Let $S$ be the sample space of all $3 \times 3$ 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 $E_2=\{A \in S : \text{sum of entries of } A \text{ is } 7\}$. If a matrix is chosen at random from $S$, then the conditional probability $P(E_1/E_2)$ equals ____.

Let S be the sample space of all 3 × 3 matrices with entries from the set {0,1}. Let the events E₁ and E₂ be given by E₁={A ∈ S : det A = 0} and E₂={A ∈ 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 ____. $S = \{3\times 3 \text{ matrices, entries } 0 \text{ or } 1\}$ $E_1: \det A = 0,\qquad E_2: \text{sum of entries} = 7$ $\text{To find: } P(E_1 \mid E_2) = \dfrac{|E_1 \cap E_2|}{|E_2|}$ $|S| = 2^9 = 512$ $|E_2| = \binom{9}{7} = \binom{9}{2} = 36$ $P(E_1 \mid E_2) = \dfrac{|E_1 \cap E_2|}{|E_2|}$ $\text{Plan: classify the 2 zeros} \Rightarrow \text{count } \det A = 0$ $\text{Case 1: two zeros in one row} \Rightarrow \text{two equal rows}$ $\det A = 0,\quad \text{count} = 3\binom{3}{2} = 9$ $\text{Case 2: two zeros in one column} \Rightarrow \text{two equal columns}$ $\det A = 0,\quad \text{count} = 3\binom{3}{2} = 9$ $A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ $\det A = 1 \neq 0$ $\text{Case 3: count} = 36 - 9 - 9 = 18\ \text{(all invertible)}$ $|E_1 \cap E_2| = 9 + 9 = 18$ $P(E_1 \mid E_2) = \dfrac{|E_1 \cap E_2|}{|E_2|} = \dfrac{18}{36} = \dfrac{1}{2}$ $\boxed{P(E_1 \mid E_2) = \dfrac{1}{2}}$