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 PE1E2 equals____

JEE Advanced · Mathematics · 32. Probability

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
$E_{1} = \{A \in S : d e t A = 0\}$ 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____

A 0.1
B 0.3
C 0.5
D 0.7

Verified Solution

Answer & Solution

Correct Answer

(C) 0.5

Step-by-step Solution

Detailed explanation

For sum to be seven, we need seven

1 s

and two

0 s

Number of different possible matrices=

= \frac{9}{7 2} = 36

Eg.

[\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}]

{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.

[\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]

Now, arrangement of 0s and

1 s = \underset{\begin{matrix} S e l e c t i o n o f \\ a c o l u m n / r o w \end{matrix}}{6} \times \underset{\begin{matrix} 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 \end{matrix}}{3} = 18

Hence, required probability

= \frac{18}{36} = \frac{1}{2}

(Note: If two zero's lies on different row/column then

|\begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{matrix}| = + 1 \neq 0

)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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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 byE1=A∈S: det A=0 andE2={A∈S: sum of entries of A is 7}.If a matrix is chosen at random from S, then the conditional probability PE1E2 equals____

Step-by-step Solution