chapter-12 -worked-out-examples-5

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  (independence)

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 { since ,  are independent }  and  are independent 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:  : Man reports a six  : Die shows a six  : Die does not show a six 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. Way (Way) Actual result : Six : Man says the truth Actual result : Not a six : Man lies 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.