Answer:
To find the probability that your distance from the road at the end of your walk is greater than 1/2, we can use a geometric probability approach.
First, let's consider the possible outcomes of your walk:
1. The distance you walk, x, can take any value in the interval [0, 1].
2. The angle θ can take any value in the interval [0, π].
Now, we want to find the probability that your distance from the road at the end of your walk is greater than 1/2. This means that you need to be outside a circle with radius 1/2 centered at the road. The area of this circle is π(1/2)^2 = π/4.
To find the probability, we'll calculate the ratio of the area outside this circle to the total possible area of outcomes (the rectangle defined by [0, 1] for x and [0, π] for θ).
The total possible area of outcomes is:
Total Area = (1 - 0) * (π - 0) = π
Now, we want to find the area outside the circle (i.e., the area where your distance from the road is greater than 1/2). This area is the difference between the total area and the area of the circle:
Area outside circle = Total Area - π/4 = π - π/4 = 3π/4
Now, we can find the probability:
Probability = (Area outside circle) / (Total Area) = (3π/4) / π = 3/4
So, the probability that your distance from the road at the end of your walk is greater than 1/2 is 3/4.
Explanation: