Probability
Probability
nta_pyq_2025_jan
Grade 12
Question:
Let $A=[a_{ij}]$ be a $2\times 2$ matrix such that $a_{ij}\in\{0,1\}$ for all $i,j$ and $P(a_{ij}=0)=P(a_{ij}=1)=\dfrac{1}{2}$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then the variance of $X$ is:
$\dfrac{3}{4}$
$\dfrac{5}{8}$
$\dfrac{3}{8}$
$\dfrac{1}{4}$
Step-by-Step Solution
Key Concept: $X=ad-bc\in\{-1,0,1\}.$ Each of the $16$ matrices is equally likely. Count outcomes for each value: $X=1$ requires $ad=1$ and $bc=0$ ($1\cdot 3=3$ matrices); same for $X=-1.$
$P(X=1)=\dfrac{1\cdot 3}{16}=\dfrac{3}{16}$ (ad=1: 1 way; bc=0: 3 ways).
$P(X=-1)=\dfrac{3}{16}$ by symmetry.
$P(X=0)=\dfrac{10}{16}.$
$E[X]=0,\ E[X^{2}]=\dfrac{3+3}{16}=\dfrac{6}{16}=\dfrac{3}{8}.$ So $\operatorname{Var}(X)=\dfrac{3}{8}.$
Correct Answer: 3