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 representing the events A and B in the experiment of rolling two fair dice, we need to consider the possible outcomes and the conditions for each event.
Event A: Total is less than eight
Event B: Total is even
Step 1: Calculate the sample space
The sample space is the set of all possible outcomes when rolling two fair dice. Each die has six sides, so the total number of possible outcomes is 6 * 6 = 36.
Step 2: Determine the outcomes for event A
To find the outcomes for event A, we need to consider the sums of the two dice that are less than eight. We can obtain such outcomes by looking at the pairs of numbers whose sum is less than eight. These pairs are:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5),
(2, 1), (2, 2), (2, 3), (2, 4),
(3, 1), (3, 2), (3, 3),
(4, 1), (4, 2),
(5, 1)
Step 3: Determine the outcomes for event B
To find the outcomes for event B, we need to consider the sums of the two dice that are even. We can obtain such outcomes by looking at the pairs of numbers whose sum is even. These pairs are:
(1, 1), (1, 3), (1, 5),
(2, 2), (2, 4),
(3, 1), (3, 3), (3, 5),
(4, 2), (4, 4),
(5, 1), (5, 3), (5, 5),
(6, 2), (6, 4), (6, 6)
Step 4: Calculate probabilities
To draw probabilities on the Venn diagram, we need to calculate the probabilities of each event. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
The number of favorable outcomes for event A is 15 (counting the outcomes for A mentioned in Step 2).
The number of favorable outcomes for event B is 18 (counting the outcomes for B mentioned in Step 3).
The probability of event A is 15/36 = 5/12.
The probability of event B is 18/36 = 1/2.
Step 5: Draw the Venn diagram
Start by drawing two overlapping circles to represent events A and B. Label each circle with the event it represents.
Inside the circle representing event A, write the probability 5/12.
Inside the circle representing event B, write the probability 1/2.
In the overlapping region of the circles, write the outcomes that satisfy both events A and B. In this case, the outcomes are: (1, 1) and (3, 3).
The remaining parts of the circles represent the outcomes that are exclusive to each event.
Note: Since the number of outcomes is limited, the Venn diagram may not be perfectly proportionate.