GATE 2025 CE (CE1) – Question 63

Engineering Mathematics · Probability and Statistics: Probability, conditional probability, descriptive statistics · 2 marks · Numerical answer

A one-way, single lane road has traffic that consists of 30% trucks and 70% cars. The speed of trucks (in km/h) is a uniform random variable on the interval (30, 60), and the speed of cars (in km/h) is a uniform random variable on the interval (40, 80). The speed limit on the road is 50 km/h. The percentage of vehicles that exceed the speed limit is __________________ (*rounded off to 1 decimal place*).

Note: $X$ is a uniform random variable on the interval $(\alpha, \beta)$, if its probability density function is given by

$$f(x) = \begin{cases} \frac{1}{\beta - \alpha}, & \alpha < x < \beta \\ 0, & \text{otherwise} \end{cases}$$

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Correct answer: 62.19 to 62.81

Explanation

For trucks, $P(\text{speed} > 50) = \frac{60 - 50}{60 - 30} = \frac{1}{3}$. For cars, $P(\text{speed} > 50) = \frac{80 - 50}{80 - 40} = 0.75$. Weighting by the share of each type gives $0.3 \times \frac{1}{3} + 0.7 \times 0.75 = 0.1 + 0.525 = 0.625$, so 62.5 % of the vehicles exceed the limit.