Probability
Geometric probability; conditional probability
Grade Class 12
Question:
Two positive real numbers $x$ and $y$ satisfying $x\leq1$ and $y\leq1$ are chosen at random. The probability that $x+y\leq1$, given that $x^2+y^2\geq\frac{1}{4}$, is
$\dfrac{8-\pi}{16-\pi}$
$\dfrac{4-\pi}{16-\pi}$
$\dfrac{4-\pi}{8-\pi}$
None of these
Step-by-Step Solution
Key Concept: Use conditional probability. The sample space is the unit square $[0,1]^2$. Area of event $B=(x^2+y^2\geq1/4)$ and $A=(x+y\leq1)$.
$P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{1/2-\pi/16}{1-\pi/16}=\frac{8-\pi}{16-\pi}$.
Correct Answer: 1