The total area of the rectangle is given by: \[ \text{Area of rectangle} = \text{Length} \times \text{Width} = 2 \, \text{m} \times 3 \, \text{m} = 6 \, \text{m}^2 \] The area of the circular region is: \[ \text{Area of circle} = \pi r^2 = \pi \times (0.5)^2 = \frac{\pi}{4} \, \text{m}^2 \] Thus, the winning probability is the ratio of the area of the circle to the area of the rectangle: \[ \text{Winning probability} = \frac{\text{Area of circle}}{\text{Area of rectangle}} = \frac{\frac{\pi}{4}}{6} = \frac{\pi}{24} \] Approximating \( \pi \) as \( 3.14 \): \[ \text{Winning probability} = \frac{3.14}{24} \approx \frac{11}{84} \]
The correct answer is option (A): \(\frac{11}{84}\)
Four students of class XII are given a problem to solve independently. Their respective chances of solving the problem are: \[ \frac{1}{2},\quad \frac{1}{3},\quad \frac{2}{3},\quad \frac{1}{5} \] Find the probability that at most one of them will solve the problem.
Two persons are competing for a position on the Managing Committee of an organisation. The probabilities that the first and the second person will be appointed are 0.5 and 0.6, respectively. Also, if the first person gets appointed, then the probability of introducing a waste treatment plant is 0.7, and the corresponding probability is 0.4 if the second person gets appointed.
Based on the above information, answer the following