Help with probability question

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Homework Help Overview

The discussion revolves around a probability problem involving the independence of events related to rolling a die and tossing a coin. The original poster presents two parts: one concerning a six-sided die and another regarding a fair coin.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to calculate probabilities for events A and B in both parts but expresses uncertainty about demonstrating independence mathematically.
  • Some participants question the definitions and equations related to independence, specifically whether the correct relationship is being used.
  • There is confusion regarding the probabilities of combined events, particularly in the context of the die problem.
  • Participants engage in clarifying the correct interpretation of events and their probabilities, especially in relation to the outcomes of the die and coin.

Discussion Status

The discussion is active, with participants providing guidance on definitions and clarifying misunderstandings about the independence of events. There is an ongoing exploration of the correct probabilities and their implications for independence, but no consensus has been reached on the final conclusions for both parts of the problem.

Contextual Notes

Participants note the omission of a "Relevant Equations" section in the original post, which may affect the clarity of the discussion. There is also mention of potential errors in the stated probabilities for part b, indicating that assumptions may need to be revisited.

Mark53
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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
 
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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.
 
FactChecker said:
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?
 
Mark53 said:
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'.
Mark53 said:
p(AuB)=1/4
Are you saying that the probability of rolling an even number less than 4 is 1/4?
 
haruspex said:
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
 
Mark53 said:
Sorry should say AnB=P(A) x P(B) and that P(AnB)= 1/4
You did not answer my question:
haruspex said:
Are you saying that the probability of rolling an even number less than 4 is 1/4?
 
haruspex said:
You did not answer my question:
no I am saying that it is 1/2
P(A) = 1/2
 
Mark53 said:
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?
 
haruspex said:
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
 
  • #10
Mark53 said:
1/6
Right. So is P(A&B)=P(A)P(B)?
 
  • #11
haruspex said:
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
 
  • #12
Mark53 said:
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.
 
  • #13
haruspex said:
Yes, and yes.
so this would mean that part b would be independent
 
  • #14
Checking the details of the part b answer, I think your original post has a wrong value for P(B).
 
  • #15
Mark53 said:
so this would mean that part b would be independent
Right.
 
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