Probability

A cube-shaped die has 6 faces, and the even numbers on the die are 2, 4, and 6. Therefore, the probability of rolling an even number on a single die is: Since the dice are rolled simultaneously, the probability that both dice show an even number is the product of the individual probabilities: Final Answer:

The problem involves finding the probability that a randomly chosen point within the square falls inside the shaded area formed by the intersection of two circular sectors with radius equal to the side length of the square.

Step 1: Understand the geometry of the problem.
- Let the side of the square PQRS be .
- The shaded area is formed by the intersection of two sectors of circles, each with radius and centers at and .
- The area of each sector is given by: where is the central angle of the sector. In this case, , so the area of each sector is: Step 2: Calculate the area of the shaded region.
- The shaded area is formed by the intersection of two such sectors. The total area of the sectors is , but we need to subtract the area of the overlap between the two sectors, which is . Hence, the area of the shaded region is: Step 3: Calculate the probability.
- The area of the square is , so the probability that a randomly chosen point falls inside the shaded region is: Thus, the correct answer is option (C).

Final Answer: