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

Ah, probabilities! The land of uncertainty and oh-so-many calculations. Let's see if we can add a little humor to this mathematical adventure.

P(A) - The probability that the magnitude of the difference of the two numbers is greater than 1/3. Well, let's just say that in a sea of infinite numbers between zero and one, this event is as likely as finding a unicorn riding a unicycle. Incorrect!

P(B) - The probability of at least one of the numbers being greater than 1/4. You know, it's like trying to find a needle in a haystack, but instead, you're looking for a number among the infinite grains of sand in the vast desert of possibilities. Incorrect!

P(A∩B) - The probability that events A and B both occur. Now we're talking about the intersection of unicorns and needle-finding. It's as tricky as trying to play a game of hide and seek with the invisible man. Incorrect!

P(C) - The probability that the sum of the two numbers is 1. Well, you see, when Alice and Bob come together, it's like the perfect balance of yin and yang, peanut butter, and jelly...wait, where was I? Oh, yes, the probability is correct!

P(D) - The probability that Alice's number is greater than 1/4. Ah, Alice, always trying to be above average. But in the realm of probabilities, being greater than 1/4 is like being a fire-breathing dragon in a world of meek little squirrels. Incorrect!

P(A∩D) - The probability that events A and D both occur. It's like trying to catch a rainbow and then realizing rainbows are made of unicorn farts. Highly unlikely, my friend. Incorrect!

To find the probabilities, we need to determine the areas of the regions corresponding to each event on the unit square.

Event A is defined as the magnitude of the difference of the two numbers being greater than 1/3. This means that the numbers must be more than 1/3 apart from each other. Considering the unit square, we can visualize this condition as two diagonal strips each with a width of 1/3. So the area of A is given by:

P(A) = (1 - 1/3) * (1 - 1/3) = (2/3) * (2/3) = 4/9

Event B is defined as at least one of the numbers being greater than 1/4. This means that the area of B is the complement of the region where neither number is greater than 1/4, which is a square with a side length of 1/4. So the area of B is:

P(B) = 1 - (1/4) * (1/4) = 1 - 1/16 = 15/16

To find the probability of the intersection of A and B, we need to find the common region between A and B, which is a rectangle formed by the intersection of the two diagonal strips (1/3 apart) and the square with side length 1/4. The length of the rectangle is 1/4, and the width is 1/3, so the area of A∩B is:

P(A∩B) = (1/4) * (1/3) = 1/12

Event C is defined as the sum of the two numbers being 1. The region where the sum is 1 is a diagonal line dividing the unit square. The area of C is given by the length of the diagonal line (1) divided by the area of the unit square (1):

P(C) = 1/1 = 1

Event D is defined as Alice's number being greater than 1/4. This means that the area of D is the complement of the region where Alice's number is not greater than 1/4, which is a square with side length 1/4. So the area of D is:

P(D) = 1 - (1/4) * (1/4) = 1 - 1/16 = 15/16

To find the probability of the intersection of A and D, we need to find the common region between A and D, which is a diagonal strip (1/4 wide) within the diagonal strip of A (1/3 wide). So the area of A∩D is:

P(A∩D) = (1/4) * (1/3) = 1/12

Therefore, the correct probabilities are:

P(A) = 4/9
P(B) = 15/16
P(A∩B) = 1/12
P(C) = 1
P(D) = 15/16
P(A∩D) = 1/12

To find the probabilities, we need to understand the definitions of the events and calculate the areas that correspond to each event on the unit square.

Let's go through each event and calculate their respective probabilities:

A = {The magnitude of the difference of the two numbers is greater than 1/3}
To calculate the probability of A, we need to find the area on the unit square where the magnitude of the difference between the two numbers is greater than 1/3. This area can be visualized as two triangles on opposite corners of the square, each with base 2/3 and height 1/3. Therefore, the total area is two times the area of one triangle, which is (1/2) * (2/3) * (1/3) = 1/9. Hence, P(A) = 1/9.

B = {At least one of the numbers is greater than 1/4}
To calculate the probability of B, we need to find the area on the unit square where at least one of the two numbers is greater than 1/4. To visualize this area, we can consider the complement of B, which is the area where both numbers are less than or equal to 1/4. This area forms a square with side length 1/4, which gives us an area of (1/4)^2 = 1/16. Therefore, P(B) = 1 - P(B complement) = 1 - 1/16 = 15/16.

A ∩ B = {Both A and B occur}
To find the probability of the intersection of A and B, let's consider the areas that correspond to both events. The area for A is 1/9, as we found earlier, and the area for B is 15/16, as we calculated previously. To find the intersection, we need to find the common area between the two regions. However, in this case, A and B do not have any overlap, so their intersection is empty. Therefore, P(A ∩ B) = 0.

C = {The sum of the two numbers is 1}
To calculate the probability of C, we need to find the area on the unit square where the sum of the two numbers is exactly 1. This area forms a diagonal line across the square from (0,1) to (1,0). The diagonal line divides the square into two triangles with equal areas. Therefore, the probability of C is the area of one of the triangles, which is (1/2) * 1 * 1 = 1/2. Hence, P(C) = 1/2.

D = {Alice's number is greater than 1/4}
To calculate the probability of D, we need to find the area on the unit square where Alice's number is greater than 1/4. This area forms a rectangle with dimensions 3/4 by 1. Therefore, the probability of D is the area of this rectangle, which is (3/4) * 1 = 3/4. Hence, P(D) = 3/4.

A ∩ D = {Both A and D occur}
Similar to earlier, let's consider the areas that correspond to A and D. We already found that the area for A is 1/9, and the area for D is 3/4. However, these two events do not have any overlap, so their intersection is empty. Therefore, P(A ∩ D) = 0.

To summarize:

P(A) = 1/9
P(B) = 15/16
P(A ∩ B) = 0
P(C) = 1/2
P(D) = 3/4
P(A ∩ D) = 0

4/9, 15/16, 4/18, 0, 0, 0