Analytical Reasoning Question I I would appreciate it if someone could please help me understand what is being asked here and how to approach questions like the one below.
"At the college entrance exam, a candidate is admitted according to whether he has passed or failed the test. Of the candidates who are really capable, 80 % pass the test and of the incapable, 25 % pass the test. Given that 40 % of the candidates are really capable, then the proportion of the really capable students who can pass the test to the total students who can pass is?"
 A: You don’t need any fancy notation to solve the problem.
You know that $40$% of the candidates are really capable and that $80$% of those $40$% pass the test; $0.8 \cdot 0.4 = 0.32$, so $32$% of all candidates both pass the test and are really capable. Of the remaining $60$% of the candidates, $25$% pass; $0.25 \cdot 0.6 = 0.15$; so $15$% of all candidates both pass the test and aren’t really capable. Altogether, then, $32+15=47$% of the candidates can pass the test, and the fraction of those who are really capable is $32/47$.
Alternatively, you can do as gary suggested and imagine that you’re working with a specific number of candidates. Choose the number so that all of the percentages work out to whole numbers of people; in this case $100$ works. Then you have $40$ who are really capable, of whom $32$ pass, and $60$ who aren’t capable, of whom $15$ pass anyway. Thus, $47$ pass, of whom $32$ are really capable, and the desired proportion is $32/47$. As you can see, this is just doing with specific numbers what I did with the percentages in the previous paragraph.
A: We have $$p(P|C) = 0.8$$ $$p(P|C') = 0.25$$ and $$p(C) = 0.4$$ and $$p(C') = 0.6$$
Note that $P$ denotes the event of passing and $C$ denotes the event that a person is capable. Thus $$p(P \cap C) = p(P|C)p(C) = (0.25)(0.4)$$
Also $$p(P) = p(P \cap C)+ p(P \cap C')$$
$$ = p(P|C)p(C)+ p(P|C')p(C')$$
$$ = (0.8)(0.4)+(0.25)(0.6)$$
So compute $$\frac{p(P \cap C)}{p(P)}$$
A: Assume that there are 100x students
as given 40% (40x)are capable imply that remaining 60% (60x)are incapable.
So 80% of capable pass the test means 40x*.8=32x passed which are capable ...(1)
And 25% of incapable pass the test means 60x*.25=14x passed the test which are in capable
So total pass=40x+15x=47x
so proportion of the really capable students who can pass the test to the total students who can pass is
32x/47x= 32/47 ans
