Question

It is known that 15% of people are allergic to peanuts, 10% are allergic to almonds, and 5% are allergic to both.

1. If we pick one person at random, what is the chance they are allergic to peanuts or almonds?
2. What is the chance that someone is allergic to either peanuts or almonds, but not both?
3. Suppose you pick out 5 people at random. What is the chance that exactly 2 of the 5 is allergic to peanuts but not almonds?
4. Are “allergic to peanuts” and “allergic to almonds” disjoint outcomes?

Answer 1

let A=allergic to peanuts=15%=0.15

B=allergic to almonds=10%=0.10

AB=5%=0.05

1.They are allergic to peanuts or almonds:

=A+B-AB

=0.15+0.10=0.05

=0.20

2. allergic to either peanuts or almonds, but not both:

=

=

=(0.15-0.05)+(0.10-0.05)

=0.10+0.05

=0.15

3.its a case of binomial distribution:

let us define a variable:

X= allergic to peanuts but not almonds

p=success==

p=0.15-0.05

p=0.10

q=1-p=1-0.10
q=0.90

n=10

P(X=x)=

P(X=2)=

P(X=2)=10*0.01*0.729

P(X=2)=0.0729

4.

two sets are disjoint if there intersection is 0.

but here =0.050

hence, “allergic to peanuts” and “allergic to almonds” not disjoint outcomes

please rate my answer and comment for doubts.