Question

The American Council of Education reported that 47% of college freshmen earn a degree and graduate within 5 years.
	Assume that graduation records show women make up 50% of the students who graduated within 5 years,
	but only 45% of the students who did not graduate within 5 years.
	The students who had not graduated withing 5 years either dropped out or were still working on their degrees.
	Students earn a degree and graduate within 5 years					47%
	Women student graduate within 5 years					50%
	Percentage of students who are women and did not graduate within 5 years					45%
Let:	A1	=	the student graduated within five years
	A2	=	The student did not graduate within five years
	W	=	the student is a woman
	M	=	the student is a man
	Joint	and	Marginal	Probabitilities
		A1 (in 5)	A2 (not in 5)
	Women (W)
	Men (M)
	(1)	(2)	(3)	(4)	(5)
		Prior	Conditional	Joint	Posterior
	Events	Probabilities	Probabilities	Probabilities	Probabilities
		P( Ai )	P( W | Ai )	P( Ai /\ W )	P( Ai | W )
	A1
	A2
			P( W ) =

Answer 1

Let

• A1   =   the student graduated within five years      
• A2   =   The student did not graduate within five years      
• W   =   the student is a woman      
• M   =   the student is a man

We know the following probabilities

The American Council of Education reported that 47% of college freshmen earn a degree and graduate within 5 years is same as the probability that a student graduated within five years is 0.47

the probability that a student did not graduated within five years is

Assume that graduation records show women make up 50% of the students who graduated within 5 years, is same as the conditional probability that a student is a woman given that the student graduated within 5 years is 0.50

the conditional probability that a student is a man given that the student graduated within 5 years is

graduation records show women make up only 45% of the students who did not graduate within 5 years is same as the conditional probability that a student is a woman given that the student did not graduate within 5 years is 0.45

the conditional probability that a student is a man given that the student did not graduate within 5 years is

Now we can get the following joint probabilities (using the formula for conditional probabilities)

We can get the marginal probabilities as below

We can fill the following tables

Finally we need the posterior probabilities

The final table is

Let:	A1	=	the student graduated within five years
	A2	=	The student did not graduate within five years
	W	=	the student is a woman
	M	=	the student is a man
	Joint	and	Marginal	Probabitilities
		A1 (in 5)	A2 (not in 5)
	Women (W)	0.235	0.2385	0.4735
	Men (M)	0.235	0.2915	0.5265
		0.47	0.53
	-1	-2	-3	-4	-5
		Prior	Conditional	Joint	Posterior
	Events	Probabilities	Probabilities	Probabilities	Probabilities
		P( Ai )	P( W | Ai )	P( Ai /\ W )	P( Ai | W )
	A1	0.47	0.5	0.235	0.4963
	A2	0.53	0.45	0.2385	0.5037
			P( W ) =	0.4735

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