Understanding Probability: Mutually Exclusive, Independent, And/Or Problems

[DLxX]

Probability Questions: urgent help needed

I am having a pretty hard time differentiating between the difference between these two things:

Mutually Exclusive, and Independant. When do you know which formula to use?

The same thing applies with *And* and *Or* problems.

I understand how to use the formulas, but I'm having a hard time deciding which category a problem actually goes into. Without knowing which category it goes into all my actualy knowledge is pretty much useless! Please help me, I have a math test on this tommorow.

 

[mathman]

The difference between mutually exclusive and independence is not a matter of formulas, but a matter of understanding.

Two events are mutually exclusive means if either event happens, the other cannot. In set theoretic terms it means the intersection is the empty set. Probability is not needed to describe this concept.

Two events are independent (a probability notion) if P(A and B)= P(A)*P(B).

 

[Curious3141]

Adding to what Mathman said, for mutually exclusive events A and B,

$$P(A|B) = P(B|A) = 0$$

It's just a fancy pansy way of saying the same thing.

 

[uart]

As others have said this is about understanding concepts not formulas.

The first thing I'd like to point out is that "independent" and "mutually exclusive" are not at all unconnected concepts. Basically two events are either independent or they are dependent, simple enough right. The importance of "independence" is that it greatly simplifies the associated probability problem. If two events are dependant in an arbitrary way then you certainly need more information to specify the problem and it is generally more difficult to deal with.

If two events are not independent then there are various possible types and degrees of "dependence". The occurrence of one event may perhaps make the other event more likely or perhaps it may make it less likely, and there is a range of different degrees (slight through to high) to which this increased or decreased likely-hood occurs. In such problems this is usually quantified in terms of something called "conditional probabilities". Like I said above you need more parameters to describe the problem and things are less simple when inter-dependence of events is involved.

There are a couple of exceptions though to the situation where dependence makes the problem more tricky. In extreme (or limiting) cases of dependence the problem again becomes simple. The limiting case for positive dependence (positive being where one event makes the other event more likely) is where event_A always happens when Event_B happens. In this case Event_B is just a subset of Event_A and this problem can be treated more simply than a general conditional probability problem.

The limiting case for negative dependence is where the occurrence of Event_A makes the occurrence of Event_B an impossibility and visa versa. This is called mutual exclusivity and is just an extreme case of non-independence (or dependence).

 

[floped perfect]

If two events are dependent, whatever happened in the first one has affected the outcome of the second one e.g. if cards are drawn and NOT replaced, the probability of getting 2 queens say is 4/52x3/53 whereas the probability of getting two queens in an INDPENDENT situation (e.g. if the first card was replaced) would be 4/52x4/52 i.e the second part is NOT affect the outcome of the first.

AND is multiply

OR is add.

 

[uart]

AND is multiply

But only if the two events are Independant!

OR is add.

But only if the two events are Mutually Exclusive!

That was really DLxX's problem in the first place, not knowing exactly when each "formula" was applicable.





The actual formula are as follows :

1. P(A or B) = P(A) + P(B) - P(A and B)

and

2. P(A and B) = P(B) P(A|B)


Clearly (1) reduces to "P(A or B) = P(A) + P(B)" for mutually exclusive events since P(A and B) = 0 in this case.

With the second equation the term P(A|B) means "the probability of event A given that event B has occurred", this is called the conditional (or marginal) probability. Now since P(A|B) = P(A) for independant events then the second equation reduces to "P(A and B) = P(A) P(B)" in this case.

That was the point I was trying to make in my previous post, the real significance of indepedance and multually exclusive is that they are limiting cases for which the general equations reduce to something much more simple.