Relationship Between Events A and B

Oct 13, 2025 by windailys.com 29 views

Introduction

In probability theory, the sample space is the set of all possible outcomes of an experiment. When rolling two dice simultaneously, the sample space can be quite extensive. In this article, we will explore the sample space of all possible outcomes when Abdul rolls a fair six-sided die and a fair four-sided die simultaneously. We will also define the events A and B, where A is the event that the six-sided die is three, and B is the event that Abdul rolls an even number on the four-sided die.

The Sample Space

To determine the sample space, we need to consider all possible outcomes of rolling two dice simultaneously. The six-sided die has six possible outcomes: 1, 2, 3, 4, 5, and 6. The four-sided die has four possible outcomes: 1, 2, 3, and 4. Since the two dice are rolled simultaneously, we need to consider all possible combinations of outcomes.

Six-sided Die	Four-sided Die	Outcome
1	1	(1,1)
1	2	(1,2)
1	3	(1,3)
1	4	(1,4)
2	1	(2,1)
2	2	(2,2)
2	3	(2,3)
2	4	(2,4)
3	1	(3,1)
3	2	(3,2)
3	3	(3,3)
3	4	(3,4)
4	1	(4,1)
4	2	(4,2)
4	3	(4,3)
4	4	(4,4)
5	1	(5,1)
5	2	(5,2)
5	3	(5,3)
5	4	(5,4)
6	1	(6,1)
6	2	(6,2)
6	3	(6,3)
6	4	(6,4)

There are 24 possible outcomes in the sample space.

Event A

Event A is defined as the event that the six-sided die is three. This means that the outcome of the six-sided die is 3. We can identify the outcomes that satisfy this condition by looking at the sample space.

Outcome	Six-sided Die	Four-sided Die
(3,1)	3	1
(3,2)	3	2
(3,3)	3	3
(3,4)	3	4

There are 4 outcomes that satisfy the condition of event A.

Event B

Event B is defined as the event that Abdul rolls an even number on the four-sided die. This means that the outcome of the four-sided die is 2 or 4. We can identify the outcomes that satisfy this condition by looking at the sample space.

Outcome	Six-sided Die	Four-sided Die
(1,2)	1	2
(1,4)	1	4
(2,2)	2	2
(2,4)	2	4
(3,2)	3	2
(3,4)	3	4
(4,2)	4	2
(4,4)	4	4
(5,2)	5	2
(5,4)	5	4
(6,2)	6	2
(6,4)	6	4

There are 12 outcomes that satisfy the condition of event B.

Relationship Between Events A and B

We can now examine the relationship between events A and B. We can ask questions such as: What is the probability of event A occurring given that event B has occurred? We can use the sample space to answer these questions.

Outcome	Six-sided Die	Four-sided Die	Event A	Event B
(3,1)	3	1	Yes	No
(3,2)	3	2	Yes	Yes
(3,3)	3	3	Yes	No
(3,4)	3	4	Yes	Yes
(1,2)	1	2	No	Yes
(1,4)	1	4	No	Yes
(2,2)	2	2	No	Yes
(2,4)	2	4	No	Yes
(3,2)	3	2	Yes	Yes
(3,4)	3	4	Yes	Yes
(4,2)	4	2	No	Yes
(4,4)	4	4	No	Yes
(5,2)	5	2	No	Yes
(5,4)	5	4	No	Yes
(6,2)	6	2	No	Yes
(6,4)	6	4	No	Yes

We can see that there are 4 outcomes that satisfy both events A and B.

Conclusion

In this article, we have explored the sample space of all possible outcomes when Abdul rolls a fair six-sided die and a fair four-sided die simultaneously. We have defined the events A and B, where A is the event that the six-sided die is three, and B is the event that Abdul rolls an even number on the four-sided die. We have examined the relationship between events A and B, and we have seen that there are 4 outcomes that satisfy both events.

Probability of Events A and B

We can now calculate the probability of events A and B.

The probability of event A is the number of outcomes that satisfy event A divided by the total number of outcomes in the sample space.

P(A) = 4/24 = 1/6

The probability of event B is the number of outcomes that satisfy event B divided by the total number of outcomes in the sample space.

P(B) = 12/24 = 1/2

We can also calculate the probability of event A given that event B has occurred.

P(A|B) = 4/12 = 1/3

We can also calculate the probability of event B given that event A has occurred.

P(B|A) = 4/4 = 1

Final Thoughts

In this article, we have explored the sample space of all possible outcomes when Abdul rolls a fair six-sided die and a fair four-sided die simultaneously. We have defined the events A and B, and we have examined the relationship between events A and B. We have calculated the probability of events A and B, and we have seen that the probability of event A given that event B has occurred is 1/3. We have also seen that the probability of event B given that event A has occurred is 1.