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
The sample space would consist of the following
(1,1),(1,2)...(1,6)
(2,1),(2,2)...)2,6)
..
..
(6,1).........(6,6) there are 36 of these
for your b) I don't know what you mean with your symbols. I assume you probably want the intersection and union of these sets.
A∩B would be the "intersection" of the two sets, that is, all those elements that are found in both the first set AND the second set. I see that would be {4,7,8}
AUB, A "union" B, is the set of elements that are found in either A OR B.
That would be {2,3,4,5,6,7,8,9}
To draw the Venn diagram, draw two overlapping circles.
Place the numbers 4,7, and 8 in the overlapping part, the "intersection".
Place the remaining numbers in their corresponding regions of the circles.
(a) To list the elements of the sample space, we consider the possible outcomes when a pair of dice are rolled. Each die has six sides numbered 1 to 6. When two dice are rolled, each side of the first die can be paired with each side of the second die, resulting in a total of 36 possible outcomes. We can represent these outcomes as ordered pairs, where the first number represents the outcome on the first die, and the second number represents the outcome on the second die. The sample space would consist of all the possible outcomes:
Sample space = {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), ..., (6, 6)}
(b) Let's consider the sets A = {2, 3, 4, 7, 8, 9, 10} and B = {4, 5, 6, 7, 8}.
- A' (A complement) represents the set of outcomes that are not in A. In this case, it would be all the possible outcomes in the sample space that are not present in set A. So, to find A', we need to subtract set A from the sample space.
A' = Sample space - A
A' = {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), ..., (6, 6)} - {2, 3, 4, 7, 8, 9, 10}
A' = {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), ..., (6, 6)} - {2, 3, 4, 7, 8, 9, 10}
Note: In this case, "..." indicates the continuation of the list of ordered pairs.
- A ∪ B (A union B) represents the set of outcomes that are present in either set A or set B. In other words, it is the combination of both sets.
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9, 10}
- A ∩ B (A intersection B) represents the set of outcomes that are present in both set A and set B. It is the overlapping region of both sets.
A ∩ B = {4, 7, 8}
(c) To draw the Venn diagram, you can imagine two overlapping circles representing sets A and B. Place the numbers 4, 7, and 8 in the overlapping region, as they are the common elements in both sets. Place the remaining numbers that are unique to each set in their respective regions of the circles. This will visually represent the different events:
Venn diagram:
_______
| |
A: | 2, 3, |
| 9,10 |
|_______|
_______
| |
B: | 5, 6 |
|_______|
The overlapping region contains the elements {4, 7, 8}, which represent the intersection A ∩ B. The regions outside of the overlapping portion represent the differences between A and B, and vice versa.