question_answer

If points \[x,y\] are chosen randomly from the intervals \[[0,2]\] and \[[0,1]\] respectively, then the probability that \[y\le {{x}^{2}}\] is

A) \[\frac{1}{2}\]

B) \[\frac{2}{3}\]

C) \[\frac{3}{4}\]

D) \[\frac{1}{4}\]

Correct Answer: B

Solution :

Picking two points \[x\] and \[y\] randomly from the intervals [0, 2] and [0, 1] is equivalent to picking a single point (x, y) randomly from the rectangle S shown in the adjacent figure, which has vertices at (0, 0), (2, 0), (2, 1) and (0, 1).

So, we take S as our sample space.

Now, the condition \[y\le {{x}^{2}}\] is satisfied if and only if the point (x, y) lies in the shaded region. It is the portion of the rectangle lying

Below the parabola \[y={{x}^{2}}\]

\[\therefore \]Required probability

\[a=\frac{Area of the shaded region}{Area of the rec\tan gle S}\]

Area of rectangle \[S=2\times 1=2\]

Area of shaded region

\[=\int\limits_{0}^{1}{{{x}^{2}}dx+1\times 1=\frac{1}{3}+1=\frac{4}{3}}\]

\[\therefore \]Required probability\[\frac{\frac{4}{3}}{2}=\frac{2}{3}\]