Example A.11 - Gujarati 4th Edition

[Dr. Jayanthi Sankaran]

Hi David,

Q. A. 11. In an introductory accounting class there are 500 students, of which 300 are males and 200 are females. Of these, 100 males and 60 females plan to major in accounting. A student is selected at random from this class, and it is found that this student plans to be an accounting major. What is the probability that the student is a male?

A. A. 11. Let A denote the event that the student is male and B that the student is an accounting major. Therefore, we want to find out P(A|B). From the formula of conditional probability just given, this probability can be obtained as

P(A|B) = P(AB) = 100/500
P(B) 160/500
= 0.625

Although, this example appears to be very simple, I am confused. Does the above answer imply that events A and B are not statistically independent? Specifically I don't understand the numerator 100/500...

Thanks
Jayanthi

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[ShaktiRathore]

Hi
P(A|B) is the joint probability of A and B occuring simultaneously that is student does the accounting major(B) and also is a male(A) so there are 100 out of total sample space 500 who are males(A) as well as wants to pursue acc major(B). So P(AB))=100/500.
You can also find P(AB)=P(A/B).P(B)=P(B/A).P(B) bayes theorem
P(B)=160/500 prob of acc major out of ss500,P(B/A)=100/160 prob of males given acc major is done so select males out of reduced ss of 160,cond probability reduce ss as now we need to select males out of acc major.
So P(AB)=(160/500)*(100/160)=100/500
Also P(A/B)*P(B)=P(AB)=100/500=>P(A/B)=(100/500)/(P(B)=(100/500)/(160/500)=5/8=.625
Thanks

 

[Dr. Jayanthi Sankaran]

Hi Shakti,

P(A|B) is not the joint probability of A and B occurring simultaneously - it is the conditional probability that the student is a male (A) given that he chooses to pursue accounting as a major(B) ie. P(A|B) = P(AB)/P(B).

You can also find P(AB)=P(A/B).P(B)=P(B/A).P(A) bayes theorem
P(B)=160/500 prob of acc major out of ss500,P(A/B)=100/160 prob of males given acc major is done so select males out of reduced ss of 160,cond probability reduce ss as now we need to select males out of acc major.
So P(AB)=(160/500)*(100/160)=100/500
[Also P(A/B)*P(B)=P(AB)=100/500=>P(A/B)=(100/500)/(P(B)=(100/500)/(160/500)=5/8=.625]

My question was whether events A and B are statistically independent? And they are not because for statistical independence:

P(A|B) = P(AB)/P(B) = P(A)*P(B)/P(B) = P(A)

Thanks
Jayanthi!

 

[brian.field]

Let A = Male, B = Accounting Major

Then,

P(A) = P(Male) = 300/500 = 0.600
P(A and B) = P(Male and Acct) = 100/500 = 0.200
P(B) = P(Acct) = 160/500 = 0.320

P(A and B)/P(B) = 0.20 / 0.32 = 0.625 and we know P(A) = 0.600.

Since P(A) not equal to P(A|B) = P(AB)/P(B), A and B are not independent!

Brian

 

[Dr. Jayanthi Sankaran]

Thanks, Brian - neat derivation!

Jayanthi

 

[brian.field]

The approach I used is pretty standard (for this type of question).

I might add that this is definitely an exam-type question.

Brian

 

[Dr. Jayanthi Sankaran]

Thanks, Brian!

Jayanthi

 

[ShaktiRathore]

Hi jayanti,
Yes i mistakenly put the | sign it should be P(AB) instead of P(A|B).
Yes the events are not statistically independent because P(A|B)<>P(A) which is the required condition for independence of A and B but here P(A|B)=P(AB)/P(B)<>P(A) as is observed from the following problem.
Thanks

 

[ShaktiRathore]

Also Jayanti whenever there are conditional probabilities b/w events are given it automatically means that events are not independent. Here event acc major event is influencing the no of male/female students events so they are dependent. Rolling two dies have ind outcomes which are ind events,outcome of one does not influence outcome of another. But in our example outcome of one event i.e no of males is influenced by outcome of another event acc major. P(male) changes from 300/500 to 100/160 when acc major event is taken into a/c so acc major event influence prob of male event.
Thanks

 

[Dr. Jayanthi Sankaran]

Thanks, Shakti for taking the extra mile to clarify. Wonder how it escaped me that the word 'conditional' itself means 'dependent' events!

Jayanthi

 

[brian.field]

Respectfully, I must disagree with this statement, "...whenever there are conditional probabilities b/w events are given it automatically means that events are not independent."
You can have conditional probabilities with independent events. For example, let's assume we are given the following probability distribution (below).



Then, assume we are given the conditional probability of Male given Accounting, i.e., P(M|A) = 0.50.

This does not mean that M and A are dependent. For example, P(M|A) = P(M) which proves that indeed M and A are independent.

 

[brian.field]

Conditional does NOT necessarily mean dependent!

 

[Dr. Jayanthi Sankaran]

Thanks for taking the time to clarify my doubt, Brian. I appreciate it.

Jayanthi!

 

[ShaktiRathore]

Yes
Agree with u brian conditional does not mean dependent only,conditional can also imply independent events but i forgot to attach the condition that P(A|B)<>P(B) with my statement that conditional imply dependence only when above inequality holds. conditional can also imply independence when P(A|B)=P(A) which brian has in his example.
I also said above bevause most examples give conditional probs when events are not u independence. sually independent because if the events are independent whats the use of condition in the first place. So if a question has some conditional probabilities we can immediately infer there is high chance of dependence b/w events rather than independence, we can further check and assure ourself that dependence is their by checking condition P(A|B)<>P(A). If the condition do not hold independence is implied.
Thanks

 

[Dr. Jayanthi Sankaran]

BTW what exactly is b/w?

 

[brian.field]

between