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GATE 2023 CH – Question 36

Engineering Mathematics · Probability and Statistics: Probability, conditional probability, descriptive statistics · 2 marks · Multiple choice

An exhibition was held in a hall on 15 August 2022 between 3 PM and 4 PM during which any person was allowed to enter only once. Visitors who entered before 3:40 PM exited the hall exactly after 20 minutes from their time of entry. Visitors who entered at or after 3:40 PM, exited exactly at 4 PM. The probability distribution of the arrival time of any visitor is uniform between 3 PM and 4 PM. Two persons $X$ and $Y$ entered the exhibition hall independent of each other. Which one of the following values is the probability that their visits to the exhibition overlapped with each other?

  1. $\frac{5}{9}$
  2. $\frac{4}{9}$
  3. $\frac{2}{9}$
  4. $\frac{7}{9}$

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Correct answer: (A) $\frac{5}{9}$

Explanation

Measure the time $t$ in minutes after 3 PM, uniform on $[0, 60]$. The visits do not overlap only if the earlier visitor arrives before 40 and leaves before the other arrives, that is $t_a < 40$ and $t_b > t_a + 20$ (a visitor entering at or after 40 stays until 4 PM and so overlaps with anyone who arrives later). The probability is $2 \times \frac{1}{3600}\int_0^{40}(40 - t)dt = \frac{2 \times 800}{3600} = \frac{4}{9}$. So the probability of an overlap is $1 - \frac{4}{9} = \frac{5}{9}$.