Alice and Bob each choose at random a real number between zero and one. We assume that the pair of numbers is chosen according to the uniform probability law on the unit square, so that the probability of an event is equal to its area.

We define the following events:

A = {The magnitude of the difference of the two numbers is greater than 1/3}
B = {At least one of the numbers is greater than 1/4}
C = {The sum of the two numbers is 1}
D = {Alice's number is greater than 1/4}
Find the following probabilities:

P(A)=incorrect
P(B)=incorrect
P(A∩B)=incorrect
P(C)=correct
P(D)=incorrect
P(A∩D)=incorrect

P(A) = 4/9

P(B) = 0.9375
P(A∩B)= 4/9
P(C) = 0
P(D) = 3/4
P (A∩D) = 11/36

P(A) = 4/9

P(B) = 15/16
P(A∩B)= 4/9
P(C) = 0
P(D) = 3/4
P (A∩D) = 89/288

a number is chosen at random from 1 to 50. find the probability of selecting numbers less than 23.

P(D) = 0.75

P(A) = 4/9

P(C) = 0
P(D) = 3/4

Anyone know the solution for the below?:
P(B)
P(A∩B)
P(A∩D)

P (B) = .9375

P (A∩D) = 11/36

P(A∩B) = 4/9

P(A) = 4/9

P(B) = 1-(1/16)
P(A∩B)= 4/9
P(C) = 0
P(D) = 3/4
P (A∩D) = 11/36

To find the probabilities, we need to consider the definitions of the events and analyze the given information.

Let's start with event A, which is defined as the magnitude of the difference between the two numbers being greater than 1/3. To find the probability of event A, we need to determine the area in the unit square that satisfies this condition.

Visualizing the unit square, we can see that the numbers chosen by Alice and Bob can be represented by points in this square. For the difference between the two numbers to be greater than 1/3, we can draw a region on the unit square that satisfies this condition. This region includes two triangles, one for positive differences and one for negative differences.

To calculate the probability of event A, we need to find the area of the region that satisfies the condition. The area of a single triangle can be calculated as 1/2 * base * height, where the base is 1 (the length of the unit square) and the height is 1/3. Therefore, the area of a single triangle is 1/6. Since there are two triangles, the total area is 2 * 1/6 = 1/3.

Thus, the probability of event A, P(A), is equal to the area of the region that satisfies the condition, which in this case is 1/3.

Moving on to event B, which is defined as at least one of the numbers being greater than 1/4. In this case, we can again visualize the unit square and draw a region where this condition is satisfied. This region includes three smaller squares within the unit square.

To calculate the probability of event B, we need to find the area of the region that satisfies the condition. Each of the smaller squares has a side length of 3/4, so the area of each square is (3/4)^2 = 9/16. Since there are three squares, the total area is 3 * 9/16 = 27/16.

Thus, the probability of event B, P(B), is equal to the area of the region that satisfies the condition, which in this case is 27/16.

Next, let's consider event C, which is defined as the sum of the two numbers being 1. In this case, we can visualize the unit square and draw a diagonal line from one corner to the other, representing the condition that the sum is equal to 1. We can see that the line divides the unit square into two equal triangles.

To calculate the probability of event C, we need to find the area of the region that satisfies the condition. Since the two triangles are equal in size, the area of each triangle is 1/2. Therefore, the total area is 2 * 1/2 = 1.

Thus, the probability of event C, P(C), is equal to the area of the region that satisfies the condition, which in this case is 1.

Moving on to event D, which is defined as Alice's number being greater than 1/4. Since Alice and Bob's numbers are chosen independently according to the uniform probability law, the probability of Alice's number being greater than 1/4 is equal to the probability of either Alice or Bob's number being greater than 1/4.

Therefore, the probability of event D, P(D), is equal to the probability of event B, which we calculated to be 27/16.

Finally, let's consider event A∩D, which represents the intersection of events A and D, i.e., the event where both the magnitude of the difference is greater than 1/3 and Alice's number is greater than 1/4.

To calculate the probability of event A∩D, we need to find the area of the region that satisfies both conditions. By considering the diagram, we can see that the intersection of these two events forms a smaller triangle within the larger triangles we previously calculated for events A and D.

To find the area of this smaller triangle, we can use the same formula as before: 1/2 * base * height. The base is 1/6 (from event A) and the height is 1/12 (from event D). Therefore, the area of this smaller triangle is 1/12 * 1/6 = 1/72.

Thus, the probability of event A∩D, P(A∩D), is equal to the area of the region that satisfies both conditions, which in this case is 1/72.

In summary, the correct probabilities are:

P(A) = 1/3
P(B) = 27/16
P(A∩B) = Cannot be determined with the given information.
P(C) = 1
P(D) = 27/16
P(A∩D) = 1/72