Are Coin Toss Events A, B, and C Independent?

[Rifscape]

Homework Statement

Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question.

I know for events to be independent P(A and B) = P(A) * P(B) and P(A|B) = P(A) and P(B|A) = P(B)

Though I am still having trouble conceptualizing it for this question.

A coin is to be flipped 3 times, list the possible outcomes.

Assume that each one of the outcomes has probability 1/8 of occurring, find the probability of A: Observing exactly 1 head B: Observing 1 or more heads C: Observing no heads

Are events A and B independent? Why or why not? Are events A and C independent? Why or why not? Are events B and C independent? Why or why not?

Any help is appreciated. Thank you for you help and time.

Homework Equations

P(A and B) = P(A) * P(B)
P(A|B) = P(A)
P(B|A) = P(B)

The Attempt at a Solution

Based on my current knowledge this is what I said for the first question
a.Events A and B are independent, since the probability of observing one head does not impact the probability of observing one or more heads.

 

[Ray Vickson]

Homework Statement

Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question.

I know for events to be independent P(A and B) = P(A) * P(B) and P(A|B) = P(A) and P(B|A) = P(B)

Though I am still having trouble conceptualizing it for this question.

A coin is to be flipped 3 times, list the possible outcomes.

Assume that each one of the outcomes has probability 1/8 of occurring, find the probability of A: Observing exactly 1 head B: Observing 1 or more heads C: Observing no heads

Are events A and B independent? Why or why not? Are events A and C independent? Why or why not? Are events B and C independent? Why or why not?

Any help is appreciated. Thank you for you help and time.

Homework Equations

P(A and B) = P(A) * P(B)
P(A|B) = P(A)
P(B|A) = P(B)

The Attempt at a Solution

Based on my current knowledge this is what I said for the first question
a.Events A and B are independent, since the probability of observing one head does not impact the probability of observing one or more heads.

Why not just work out the probabilities $P(A)$, $P(B)$ and $P(A\: \& \: B)$? You have the complete sample space, so that ought to be easy enough. Besides, that is the type of thing you need to practice in order to learn the subject properly.

On the other hand, you could try to come at it obliquely. I hope you realize that $A$ and $B$ are independent if, and only if $A^c$ and $B$ are independent, or if and only if $A$ and $B^c$ are independent. (Here, $E^c$ denotes the complement of an event $E$.) So, do you think that $P(A)$ and $P(A|B^c)$ are the same? Is $P(A)$ the same as $P(A|C)$?