Example: 9  
For two events and , we are given the following information:
(a) Determine whether the two events are or independent, or neither.
(b) Find . The notation stands for the complementary event of event , i.e., occurs if does not occur.

Solution: 9(a)  
Since and are the same, this means that and are independent events. Hence they are obviously not .
We therefore obtain

Solution: 9(b)  
Now, if and are independent events, then so are and (and in fact ( and ) and . This fact should be intuitively obvious but let us justify it with some rigor. We have
Similar proofs follow for the other two pairs.
Returning to the question, we see that since and are independent, we have

Example: 10  
In this example, we’ll get a taste of what is known as inverse probability, which we’ll of course be discussing in detail in a later section.
A man is known to speak the truth out of times. He throws a die and tells you that he obtained a six. What is the chance of it being really a six ?

Solution: 10  
The problem here is that some event has already taken place, i.e., the throwing of a die. What we are required to do is find out how truthful the result being told to us is, given that the person reporting the result is not exactly a saint!
So, we first let , , denote the following events:
Observe that we are required to find , i.e., the probability of the die actually showing a six given that the man is reporting so.
Since we already know a bit about conditional probabilities, let us try to follow that route of manipulation.
Now, is the event that the man reports a six when actually he obtains a six. The probability of this event will simply be
You are urged to carefully understand how we arrived at this.
Next, we try to find , i.e., the probability of the man reporting a six. This can happen in two mutually exclusive ways.
Thus,
Finally,
It should be remarked that a lot of readers might not understand fully the underlying idea of this problem. They need not despair since inverse probabilities will be dealt with in much detail later.
