JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Probability

The region is bounded by the y-axis ($x=0$), the line $2y+x=6$, and the line $5x-6y=30$.

Find the vertices: The lines $2y+x=6$ and $5x-6y=30$ intersect at $B=(6,0)$. The line $2y+x=6$ meets the y-axis at $C=(0,3)$. The line $5x-6y=30$ meets the y-axis at $A=(0,-5)$.

The triangular region $ABC$ has area $\frac{1}{2}\times 6\times 8=24$.

The region where $y<1$ is bounded by $y=1$, which intersects $2y+x=6$ at $D=(4,1)$ and $5x-6y=30$ at $E=(\frac{36}{5},1)$.

The area where $y\ge 1$ (triangle $BDE$) is $\frac{1}{2}\times 2\times 4=4$.

Required probability $=1-\frac{4}{24}=1-\frac{1}{6}=\frac{5}{6}$.