Independent vs Mutually Exclusive

[lovemake1]

Homework Statement

If a question states event A and B are "not" independent, does it mean that they are Mutually Exclusive?
My brain is having hard time accepting that if they are not dependent, then they don't neccesarily have to be 'dependent'. Kinda like if its not hot, then its cold. it can still be warm.
any help clearing this confusion is greatly appreciated.

This is introductory statistics, so nothing crazy has been introduced.

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

If a question states event A and B are "not" independent, does it mean that they are Mutually Exclusive?

No.

Events A and B are independent if the outcome of one does not effect the outcome of the other... P(A|B)=P(A)

Events are mutually exclusive if having one event means you cannot have the other... P(A|B)=0

Being mutually exclusive is one way that events can be dependent ... but not the only way.
See HallsofIvy (below).

 

[HallsofIvy]

For example, suppose you roll a single die. Event A is "you roll a number larger than 3". Event B is "you roll and even number" We can write event A as {4, 5, 6}. We can write event B as {2, 4, 6}. The probabilities of both event A and event B are 3/6= 1/2. The numbers that are both "larger than 3 and even" are {2, 6} so the probability that "you roll a number than is both larger than 3 and even" is 2/6= 1/3. That is NOT (1/2)(1/2)= 1/4 so these events are not independent. But it is not 0 so they are not "mutually excusive"

 

[jbunniii]

Homework Statement

If a question states event A and B are "not" independent, does it mean that they are Mutually Exclusive?

It's interesting to note that the converse of this statement IS true. If two events are mutually exclusive, then they cannot be independent unless one of them has probability zero. This follows immediately from the definitions of mutually exclusive: $P(A \cap B) = 0$ and of independent: $P(A \cap B) = P(A) P(B)$.

 

[Simon Bridge]

Just to clarify jbunniii:
If two events A and B are mutually exclusive, then they are not independent - but it is not true to say that "if A and B are not independent, it means that they are Mutually Exclusive".All dogs are animals but not all animals are dogs.