Alice and Bob each choose at random a number between zero and two. Consider the events events
A: The magnitude of the difference of the two numbers is greater than 1/4. Event B: At least one
of the numbers is greater than 1/4. Event C: Alice’s number is greater than 1/4. Use a graph to
find the probabilities P(B) and P(A ∩ C).
To find the probabilities P(B) and P(A ∩ C), we need to first visualize the sample space and the events A, B, and C on a graph.
Let's set up a Cartesian plane where the x-axis represents Alice's number and the y-axis represents Bob's number. Both Alice and Bob can choose any number between zero and two.
Now let's plot the region on the graph where at least one of the numbers is greater than 1/4. This region is represented by the shaded area above the line y = 1/4 and to the right of the line x = 1/4. This represents event B.
To find P(B), we need to find the area of the shaded region and divide it by the total area of the sample space.
The total area of the sample space is a square with side length 2, so the total area is 2*2 = 4.
To find the area of the shaded region, we can split it into two triangles and a rectangle. The area of each triangle is (1/2) * (2 - 1/4) * (2 - 1/4) = (7/32). The area of the rectangle is (1/4) * (2 - 1/4) = (3/8).
So, the area of the shaded region is 2 * (7/32) + (3/8) = (23/32).
Therefore, P(B) = Area of the shaded region / Total area = (23/32) / 4 = 23/128.
Next, let's plot the region on the graph where the magnitude of the difference between the two numbers is greater than 1/4. This region is represented by the shaded area above the lines y = x + 1/4 and y = x - 1/4. This represents event A.
To find P(A ∩ C), we need to find the area of the intersection between the shaded region for event A and the region where Alice's number is greater than 1/4. This represents event C.
The area of the shaded region for event A is a triangle with side length 2 and height 1/2. So the area is (1/2) * 2 * (1/2) = 1/2.
The intersection with event C occurs in the region above the line y = x - 1/4, where Alice's number is greater than 1/4, and below the line y = 2. This region is a trapezoid with bases of length 1/4 and 1, and a height of 1/2. So the area of the intersection with event C is (1/2) * (1/4 + 1) * (1/2) = 5/16.
Therefore, P(A ∩ C) = Area of the intersection / Total area = (5/16) / 4 = 5/64.
So, the probability P(B) is 23/128 and the probability P(A ∩ C) is 5/64.