Suppose stocks of companies A, B, and C are popular among investors. Suppose $18$% of the investors own A stocks, $49$% own B stocks, $32$% own C stocks, $5$% own all three stocks, $8$% own A&B stocks, $10$% own B&C stocks and $12$% own C&A stocks.
(a) What proportion of the investors own stocks of only one of these companies?
(b) What proportion of the investors do not invest in any of these three companies?
For (a) would it be A&B plus B&C plus C&A = $30$%, then $100$% - $30$% = $70$%?
For (b) would it be $18$ + $49$ + $32$ = $99$, then $100$ - $99$ = $1$%?
You can solve this by using sets (three set Venn Diagram), but basically intersections and unions should suffice. I used a set diagram to obtain this visually, I will let someone to fix my poorly written math.
Let be I the set of all investors (Universe)
I=100%
A∩B∩C=5% own all three stocks A,B and C.
A∩B=8% own A and B stocks only.
We can deduce that A∩B-A∩B∩C=3%.
B∩C=10% own B and C stocks.
We can deduce that B∩C-A∩B∩C=5%.
C∩A=12% own C and A stocks.
C∩A-A∩B∩C=7%.
To solve a), you need to single out all one company investors:
A=18%, single out pure A investor is A-A∩B-C∩A+A∩B∩C=18%-8%-12%+5%=3%
B=49%, single out pure B investor is B-A∩B-B∩C+A∩B∩C=49%-8%-10%+5%=36%
C=32%, single out pure C investor is C-C∩B-C∩A+A∩B∩C=32%-10%-12%+5%=15%
So the proportion of investors that own only one of those companies is: 3%+36%+15% = 54%
To solve b), you subtract I from all the intersections: 100%-3%-36%-15%-3%-7%-5%-5% = 26%.
(Someone help me edit this mess, thanks!)