Geometric Probability of Absolute Difference Between Two Uniform Variables

Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space [0,60] is less than or equal to a. If P(A)=11/36, then a is equal to

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Question type: Numerical
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 19 September 2026

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Why is the area of the two excluded corners equal to

(60 - a)^{2}

Each corner is an isosceles right triangle with base and height equal to $(60 - a)$ . The area of one triangle is $\frac{1}{2} (60 - a)^{2}$ . Two such symmetric triangles combine to give an area of $(60 - a)^{2}$ .

Answer

Accepted answer: 10

The value of a is 10.

Why each option works or fails

Step-by-step solution

Given: Two numbers $x, y \in [0, 60]$ are chosen independently at random. Event $A$ is $∣ x - y ∣ \leq a$ . The probability is given as $P (A) = \frac{11}{36}$ .
Goal: Find the value of $a$ such that $P (A) = \frac{11}{36}$ .
Approach: Represent the sample space as a square of side $60$ in the $x y$ -plane with area $60^{2} = 3600$ . The condition $∣ x - y ∣ \leq a$ corresponds to the region between the lines $y = x - a$ and $y = x + a$ . The unshaded/complementary region consists of two right-angled isosceles triangles at the top-left and bottom-right corners, each with legs of length $(60 - a)$ . Thus, $P (A) = 1 - \frac{(60 - a)^{2}}{60^{2}}$ . Set this equal to $\frac{11}{36}$ and solve for $a$ .
Execute: Set up the equation: $1 - \frac{(60 - a)^{2}}{3600} = \frac{11}{36}$ $\frac{(60 - a)^{2}}{3600} = 1 - \frac{11}{36} = \frac{25}{36}$ $(60 - a)^{2} = 3600 \times \frac{25}{36} = 2500$ Since $a \in [0, 60]$ , $60 - a > 0$ , taking the square root gives: $60 - a = 50 ⟹ a = 10$
Verify: For $a = 10$ , corner triangles have leg $60 - 10 = 50$ . Total corner area is $50^{2} = 2500$ . Shaded area is $3600 - 2500 = 1100$ . Probability is $\frac{1100}{3600} = \frac{11}{36}$ , which matches the given value exactly.

Geometric Probability of Absolute Difference Between Two Uniform Variables

What feels right?

The value of a is 10.

Why each option works or fails

Solution

01Given

02Goal

03Approach

04Execute

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Geometric Probability of Absolute Difference Between Two Uniform Variables

The value of a is 10.

01Given

02Goal

03Approach

04Execute

✓Verify

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