Answer: a) 65.75% and b) 13.7%
Step-by-step explanation:
Since we have given that
Number of people who are allergic to dairy products = n(A) = 71
Number of people who are allergic to pollen = n(B) = 98
Number of people who are allergic to animal dander = n(C) = 96
Number of people who are allergic to all n(A∩B∩C) = 33
Number of people who are allergic only to pollen = 30
Number of people who are allergic only to dairy product = 10
Number of people who allergic to dairy products and pollen n(A∩B)= 43
Now, we first find a, c, d, e from the Venn Diagram:
Since n(A∩B)= 43
![a+b=43\\\\a+33=43\\\\a=43-33\\\\a=10](https://tex.z-dn.net/?f=a%2Bb%3D43%5C%5C%5C%5Ca%2B33%3D43%5C%5C%5C%5Ca%3D43-33%5C%5C%5C%5Ca%3D10)
For Diary product,
![10+a+b+c=71\\\\a+b+c=61\\\\10+33+c=61\\\\c=18](https://tex.z-dn.net/?f=10%2Ba%2Bb%2Bc%3D71%5C%5C%5C%5Ca%2Bb%2Bc%3D61%5C%5C%5C%5C10%2B33%2Bc%3D61%5C%5C%5C%5Cc%3D18)
For pollen,
![a+30+b+e=98\\\\10+30+33+e=98\\\\e=98-73=25](https://tex.z-dn.net/?f=a%2B30%2Bb%2Be%3D98%5C%5C%5C%5C10%2B30%2B33%2Be%3D98%5C%5C%5C%5Ce%3D98-73%3D25)
For animal dander,
![c+b+d+e=96\\\\18+33+25+d=96\\\\d=96-76=20](https://tex.z-dn.net/?f=c%2Bb%2Bd%2Be%3D96%5C%5C%5C%5C18%2B33%2B25%2Bd%3D96%5C%5C%5C%5Cd%3D96-76%3D20)
So, Total allergic persons = 10+18+20+25+30+10+33=146
So, (a) What percent of Dr. Hawk's patients are allergic to animal dander?
Percentage that patients are allergic to animal dander is given by
![\dfrac{96}{146}\times 100=65.75\%](https://tex.z-dn.net/?f=%5Cdfrac%7B96%7D%7B146%7D%5Ctimes%20100%3D65.75%5C%25)
(b) What percent of Dr. Hawk's patients are allergic only to animal dander?
Percentage that patients are allergic to only animal dander is given by
![\dfrac{20}{146}\times =13.7\%](https://tex.z-dn.net/?f=%5Cdfrac%7B20%7D%7B146%7D%5Ctimes%20%3D13.7%5C%25)
Hence, a) 65.75%
b) 13.7%