Step 1: Identify the Total Region (R)
The given condition is x^2 + y^2 < 4, which represents the interior of a circle centered at the origin (0,0) with a radius of 2 (since r2=4⟹r=2).
We are given that the region R is restricted to the first quadrant (x > 0, y > 0). Therefore, region R represents a quadrant of the circle (one-fourth of the entire circle).
Step 2: Identify the Favorable Region (r > s)
The point P=(r,s) represents any coordinate (x,y) within region R. We need to find the probability that the x-coordinate is greater than the y-coordinate, i.e., x > y.
To find this region, consider the boundary line:
y=x
The line y=x passes through the origin and makes an angle of 45∘ with both the x-axis and the y-axis. It divides the first quadrant exactly into two equal halves:
The region below the line, where x > y (or r > s).
The region above the line, where y > x (or s > r).
Step 3: Calculate the Favorable Area
Because the line y=x is a line of symmetry for the quadrant of the circle, it divides the area of region R into two perfectly equal sectors.
Step 4: Calculate the Probability
Using the formula for geometrical probability:
Probability=Total AreaFavorable Area
Probability=π2π=21
Conclusion
The probability that r > s is 1/2.
Therefore, the correct option is C.
