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
[tex]a+b=43\\\\a+33=43\\\\a=43-33\\\\a=10[/tex]
For Diary product,
[tex]10+a+b+c=71\\\\a+b+c=61\\\\10+33+c=61\\\\c=18[/tex]
For pollen,
[tex]a+30+b+e=98\\\\10+30+33+e=98\\\\e=98-73=25[/tex]
For animal dander,
[tex]c+b+d+e=96\\\\18+33+25+d=96\\\\d=96-76=20[/tex]
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
[tex]\dfrac{96}{146}\times 100=65.75\%[/tex]
(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
[tex]\dfrac{20}{146}\times =13.7\%[/tex]
Hence, a) 65.75%
b) 13.7%
