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The formulation of conditional probability assumes the conditioning event C is well defined. Sometimes there are subtle difficulties. It may not be entirelyclear from the problem description what the conditioning event is. This is usually due to some ambiguity or misunderstanding of the information provided.

What is the conditioning event?

Five equally qualified candidates for a job, Jim, Paul, Richard, Barry, and Evan, are identified on the basis of interviews and told that they are finalists. Three ofthese are to be selected at random, with results to be posted the next day. One of them, Jim, has a friend in the personnel office. Jim asks the friend to tell him the name ofone of those selected (other than himself). The friend tells Jim that Richard has been selected. Jim analyzes the problem as follows.


Let A i , 1 i 5 be the event the i th of these is hired ( A 1 is the event Jim is hired, A 3 is the event Richard is hired, etc.). Now P ( A i ) (for each i ) is the probability that finalist i is in one of the combinations of three from five. Thus, Jim's probability of being hired, before receiving the information about Richard, is

P ( A 1 ) = 1 × C ( 4 , 2 ) C ( 5 , 3 ) = 6 10 = P ( A i ) , 1 i 5

The information that Richard is one of those hired is information that the event A 3 has occurred. Also, for any pair i j the number of combinations of three from five including these two is just the number of ways of picking one from the remaining three.Hence,

P ( A 1 A 3 ) = C ( 3 , 1 ) C ( 5 , 3 ) = 3 10 = P ( A i A j ) , i j

The conditional probability

P ( A 1 | A 3 ) = P ( A 1 A 3 ) P ( A 3 ) = 3 / 10 6 / 10 = 1 / 2

This is consistent with the fact that if Jim knows that Richard is hired, then there are two to be selected from the four remaining finalists, so that

P ( A 1 | A 3 ) = 1 × C ( 3 , 1 ) C ( 4 , 2 ) = 3 6 = 1 / 2
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Although this solution seems straightforward, it has been challenged as being incomplete. Many feel that there must be information about how the friend chose to name Richard. Many would make an assumption somewhat as follows. Thefriend took the three names selected: if Jim was one of them, Jim's name was removed and an equally likely choice among the other two was made; otherwise, the friendselected on an equally likely basis one of the three to be hired. Under this assumption, the information assumed is an event B 3 which is not the same as A 3 . In fact, computation (see Example 5, below) shows

P ( A 1 | B 3 ) = 6 10 = P ( A 1 ) P ( A 1 | A 3 )

Both results are mathematically correct. The difference is in the conditioning event, which corresponds to the difference in the information given (or assumed).

Some properties

In addition to its properties as a probability measure, conditional probability has special properties which are consequences of the way it is related to the original probability measure P ( ) . The following are easily derived from the definition of conditional probability and basic properties of the prior probabilitymeasure, and prove useful in a variety of problem situations.

(CP1) Product rule If P ( A B C D ) > 0 , then P ( A B C D ) = P ( A ) P ( B | A ) P ( C | A B ) P ( D | A B C ) .


The defining expression may be written in product form: P ( A B ) = P ( A ) P ( B | A ) . Likewise

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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