Referring to the experiment of rolling two fair dice, let A be the event that the total is less

than eight and let B be the event that the total is even. Draw a Venn diagram corresponding
to this experiment and events A & B (using probabilities).
1

To draw a Venn diagram for events A and B in the experiment of rolling two fair dice, we need to understand the sample space and the events A and B.

The sample space for rolling two fair dice is the set of all possible outcomes when the two dice are rolled simultaneously. Since each die has six sides numbered 1 to 6, there are a total of 6*6 = 36 possible outcomes.

Event A represents the sum of the numbers on the two dice being less than eight. To determine the event A, we need to find all the outcomes where the sum of the numbers is less than eight. These outcomes are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (5,1), (5,2), (6,1). So, there are 21 outcomes where the sum is less than eight.

Event B represents the sum of the numbers on the two dice being even. To determine the event B, we need to find all the outcomes where the sum of the numbers is even. These outcomes are: (1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (5,1), (5,3), (5,5), (6,2), (6,4), (6,6). So, there are 16 outcomes where the sum is even.

Now, let's draw the Venn diagram:

1. Draw a rectangle to represent the sample space. Label it S and write the number 36 to represent the total number of outcomes.

2. Inside the rectangle, draw two overlapping circles to represent events A and B. Label one circle as A and the other as B.

3. In the region corresponding to event A, write the number 21 to represent the outcomes where the sum is less than eight.

4. In the region corresponding to event B, write the number 16 to represent the outcomes where the sum is even.

5. In the region where the two circles overlap, write the number to represent the outcomes that satisfy both events A and B. In this case, it is 10 as (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5), (6,2) are the common outcomes.

Note: The counts I mentioned above are probabilities, as the probability of each outcome is 1/36 since each die is fair.

This completes the Venn diagram for events A and B in the experiment of rolling two fair dice, where event A represents the total sum less than 8, and event B represents an even total.