) In an experiment, a pair of dice is rolled and the total number of points observed.

(a) List the elements of the sample space
(b) If A = { 2, 3, 4, 7, 8, 9, 10} and B = {4, 5, 6, 7, 8} list the outcomes which comprise each of the following events and also express the events in words: A¢, A È B, and A Ç B.
(c) Use Venn diagrams to show the different events.

(a) To determine the elements of the sample space, we need to find all possible outcomes when rolling a pair of dice. Each die has six sides, numbered 1 to 6, so the total number of outcomes is 6 x 6 = 36. The sample space consists of all the possible pairs of numbers that can appear on the two dice. We can list them as follows:

Sample space = {(1, 1), (1, 2), (1, 3), ..., (1, 6), (2, 1), (2, 2), ..., (6, 5), (6, 6)}

(b) Let's analyze the events using the given sets A and B:

A = {2, 3, 4, 7, 8, 9, 10}
B = {4, 5, 6, 7, 8}

A¢ (complement of A) refers to the outcomes that are not in A. So, we need to find the outcomes that are not in set A. It is important to note that the total number of points observed from rolling two dice can range from 2 to 12.

A¢ = {1, 5, 6, 11, 12}

A È B (union of A and B) refers to the outcomes that are in either set A or set B or both. We need to combine the elements from both sets without duplication.

A È B = {2, 3, 4, 5, 6, 7, 8, 9, 10}

A Ç B (intersection of A and B) refers to the outcomes that are in both set A and set B. We need to find the common elements between both sets.

A Ç B = {4, 7, 8}

(c) To visualize the different events using Venn diagrams, we can draw two overlapping circles representing sets A and B. Set A is represented by one circle, and set B is represented by another circle overlapping with the first one. Any common elements will be shown in the overlapped region.

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In the overlapping region of the circles, we will write the common elements: 4, 7, and 8. These represent the intersection A Ç B.

To represent the complement of A, A¢, we can draw a rectangle that encloses both circles. In this rectangle, we will write the elements that are not in set A: 1, 5, 6, 11, and 12.

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