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GATE 2023 CE (CE2) – Question 13

Engineering Mathematics · Probability and Statistics: Probability, conditional probability, descriptive statistics · 1 mark · Multiple choice

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Further, even- and odd-numbered vehicles are operated on even- and odd-numbered dates, respectively. If 100 vehicles are chosen at random on an even-numbered date, the number of vehicles expected without pollution clearance certificate is ______.

  1. 15
  2. 30
  3. 50
  4. 70

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Correct answer: (B) 30

Explanation

**Step 1: which vehicles run on an even date?** The registration numbers are sequential, 1000 to 1999, so exactly half (500) are even and half are odd. On an even-numbered date only the even-numbered vehicles run.

**Step 2: certificates.** Nothing in the problem links the certificate status to the parity of the number, so we assume it is independent of it. Then 30% of the even-numbered vehicles are also without a certificate.

**Step 3: expected number in the sample.** Choosing 100 vehicles at random from those on the road:
$$E=100\times0.30=\mathbf{30}.$$

Official GATE 2023 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2023/CE2_ANS_GATE2023.pdf