Independent events (for three events)

[woody_woodpecker]

For example, if we roll a dice twice.
First event is getting a 3 in the first roll. (A)
Second event is getting a 4 in the second roll. (B)
Third event is getting 7 for total two rolls. (C)
How can we determine all three events are independent?

p(A) = 1/6
p(B) = 1/6
p(c) = (3,4)(4,3)(2,5)(5,2)(1,6)(6,1)
= 6/36
= 1/6

p(AB) = p(A)p(B)
? = 1/36

i'm not sure how to find the left side, p(AB).
same for the below parts.

p(AC) = p(A)p(C)
p(BC) = p(B)p(C)

p(ABC) = p(A)p(B)p(C)
? = (1/6)^3
= 1/216 -----> is this correct?

any help,please.
thanks!

 

[Loren]

Your question leaves me wondering what the situation truly is.

>For example, if we roll a dice twice.
First event is getting a 3 in the first roll. (A)
Second event is getting a 4 in the second roll. (B)
Third event is getting 7 for total two rolls. (C)

The word "dice" is plural. Do you mean "roll a die twice" or are you rolling more than one?
If you roll a die twice, it is my understanding that you have two events. I don't understand the "third event". Maybe you mean the "outcome".
Your question "How can we determine all three events are independent?" also leaves me wondering. I believe that you determine whether or not events are independent or dependent based on the circumstances. For me to decide in this particular case, I would ask myself "does the die remember the result of the first event and try to duplicate this or avoid duplicating it?" If I thought the outcome of the second event depended on the outcome of the first event, then I would determine that the two events were dependent. Otherwise they are independent.

 

[woody_woodpecker]

i'm sorry if my understanding of the question that makes it hard to understand.

the original question is,

consider rolling a die twice. Let A be the event that the first roll is a 3. Let B be the event that the second roll is a 4 and let C be the event that the total of the two rolls is 7. Are A, B and C independent events?

how can we justify that?

 

[Loren]

In reading the question, I at first assumed that all three events were either dependent or independent. Then, I noted that the word "all" is not included. If I were to answer the question, I would indicate that I think the three events are not all the same, regarding being dependent or independent. I would then explain what I think each event is (dependent or independent) and why I think that.