Help with probability question

[Mark53]

Homework Statement

(a) Suppose a fair six-sided die is rolled once. Let A be the event that an even face occurs and B be the event that a face less than 4 occurs. Are the events A and B independent? Show this mathematically.

(b) A fair coin is tossed three times. Let A be the event that the first toss is a Head and let B be the event all three are the same. Are the events A and B independent? Show this mathematically

The Attempt at a Solution

[/B]
a)
I know that:
P(A) = 1/2
P(B) = 1/2
but I don't know how to show that they are independent

b)
I know that P(A)=1/2 , P(B) = 1/8
but don't know how to show that they are independent

any help would be much appreciated

 

[FactChecker]

You omitted the "Relevant Equations" section. What is your definition of independent? It should be an equation that you can either prove to be true or false in each part of the question.

 

[Mark53]

You omitted the "Relevant Equations" section. What is your definition of independent? It should be an equation that you can either prove to be true or false in each part of the question.

p(AuB)=P(A) x P(B)

p(AuB)=1/4

1/4=1/2 x 1/2
1/4=1/4

therefore they are independent

Would this now be correct?

 

[haruspex]

p(AuB)=P(A) x P(B)

I assume you mean that a test for independence is that P(A&B)=P(A)P(B). AuB would mean 'or'.

p(AuB)=1/4

Are you saying that the probability of rolling an even number less than 4 is 1/4?

 

[Mark53]

I assume you mean that a test for independence is that P(A&B)=P(A)P(B). AuB would mean 'or'.

Are you saying that the probability of rolling an even number less than 4 is 1/4?

Sorry should say AnB=P(A) x P(B) and that P(AnB)= 1/4

 

[haruspex]

Sorry should say AnB=P(A) x P(B) and that P(AnB)= 1/4

You did not answer my question:

Are you saying that the probability of rolling an even number less than 4 is 1/4?

 

[Mark53]

You did not answer my question:

no I am saying that it is 1/2
P(A) = 1/2

 

[haruspex]

no I am saying that it is 1/2
P(A) = 1/2

No, event A is rolling an even number. A&B is rolling an even number less than four. What is the probability of that?

 

[Mark53]

No, event A is rolling an even number. A&B is rolling an even number less than four. What is the probability of that?

1/6

 

[haruspex]

1/6

Right. So is P(A&B)=P(A)P(B)?

 

[Mark53]

Right. So is P(A&B)=P(A)P(B)?

this means that they are not equal which means they are not independent

does this mean for part b) that P(A&B)=1/8

 

[haruspex]

this means that they are not equal which means they are not independent

does this mean for part b) that P(A&B)=1/8

Yes, and yes.

 

[Mark53]

Yes, and yes.

so this would mean that part b would be independent

 

[FactChecker]

Checking the details of the part b answer, I think your original post has a wrong value for P(B).

 

[haruspex]

so this would mean that part b would be independent

Right.