<p>Two friends Shubham and Ankur decided to discuss Probability over phone. At a random moment within period of 20 minutes, 'Shubham' telephones 'Ankur', waits for 2 minutes and then puts down the receiver. During the same 20 minutes 'Ankur' arrives home at a random moment, stays for 5 minutes and then leaves. Find the probability that the two will discuss the probability over phone.</p>
Step-by-Step Solution
Key Concept: Model this as a geometric probability problem on a 2D plane where x = Shubham's arrival time and y = Ankur's arrival time (both in [0,20]). The two friends connect if their time intervals overlap: Shubham is available during [x, x+2] and Ankur is available during [y, y+5], so they meet when |x - y| ≤ 5 and the intervals [x, x+2] and [y, y+5] intersect, which simplifies to the condition y ≤ x + 2 and x ≤ y + 5, or equivalently x - 5 ≤ y ≤ x + 2.
<p><strong>Step 1: Set up the coordinate system</strong></p><p>Let x = time Shubham calls (in minutes, 0 ≤ x ≤ 20)</p><p>Let y = time Ankur arrives home (in minutes, 0 ≤ y ≤ 20)</p><p>Total sample space = 20 × 20 = 400 (square region)</p><p><strong>Step 2: Determine the overlap condition</strong></p><p>Shubham is available during interval [x, x+2]</p><p>Ankur is available during interval [y, y+5]</p><p>These intervals overlap if and only if: x ≤ y + 5 AND y ≤ x + 2</p><p>This gives us: x - 5 ≤ y ≤ x + 2</p><p><strong>Step 3: Find the favorable region</strong></p><p>The favorable region is bounded by:</p><p>• Line 1: y = x + 2 (upper boundary)</p><p>• Line 2: y = x - 5 (lower boundary)</p><p>• Square: 0 ≤ x ≤ 20, 0 ≤ y ≤ 20</p><p><strong>Step 4: Calculate the area using complementary approach</strong></p><p>It's easier to find the area where they DON'T meet:</p><p>• Region above y = x + 2: Triangle with vertices (0,2), (0,20), (18,20) → Area = ½ × 18 × 18 = 162</p><p>• Region below y = x - 5: Triangle with vertices (5,0), (20,0), (20,15) → Area = ½ × 15 × 15 = 112.5</p><p>Unfavorable area = 162 + 112.5 = 274.5</p><p>Favorable area = 400 - 274.5 = 125.5</p><p><strong>Step 5: Calculate probability</strong></p><p>Probability = 125.5/400 = 251/800</p><p>∴ Answer: <strong>251/800 or 0.31375</strong></p>
Correct Answer: 251