Question

Conditional Probability

total A survey asked 300 Adults and Children which of three cereals they would like to have for breakfast. Use the results in the table to answer the given problems. [You do not have to simplify.]

.	Bran Flakes	Frosted flakes	Cocoa puffs	Total
Children	10	40	150	200
adults	70	25	5	100
totals	80	65	155	300

[1] Find the probability that a person prefers Frosted Flakes

[2] Find the probability that a Child does NOT want Bran Flakes.

[3] Find the probability that someone prefers Frosted Flakes given that the person is an Adult.

[4] Find the probability that a person does NOT prefer Bran Flakes.

[5] Find the probability that an Adult prefers Frosted Flakes or Cocoa Puffs.

Answer 1

Solution:

Let us define some events as follows:

A : A person prefers Bran flakes

B : A person prefers Frosted flakes

C : A person prefers Cocoa Puffs

D : Person is a child

E : Person is an adult

1) We have to find the probability that a person prefers Frosted Flakes.

i.e. We have to find P(B).

P(B) = number of persons who prefer Frosted flakes/total number of persons

number of persons who prefer Frosted flakes = 65

Total number of persons = 300

Hence, P(B) = 65/300 = 13/60

The probability that a person prefers Frosted Flakes is 13/60.

2) We have to find the probability that a Child does NOT want Bran Flakes.

P(Child does not want Bran Flakes) = 1 - P(Child want Bran Flakes).

P(Child want Bran Flakes) = Number of children who prefer Bran Flakes/total number of persons

Number of children who prefer Bran Flakes = 10

Total number of persons = 300

Hence, P(Child want Bran Flakes) = 10/300

Hence, (Child does not want Bran Flakes) = 1 - (10/300) = 290/300 = 29/30

The probability that a Child does NOT want Bran Flakes is 29/30.

3) We have to find the probability that someone prefers Frosted Flakes given that the person is an Adult.

i.e. We have to find P(B|E).

Conditional probability of B given E is given by,

P(B ∩ E) = Probability that someone prefers Frosted flakes and also he is an adult.

P(B ∩ E) = number of adult persons who prefer Frosted flakes/total number of persons

number of adult persons who prefer Frosted flakes = 25

Total number of persons = 300

Hence, P(B ∩ E) = 25/300

P(E) = number of adults/total number of persons

Number of adults = 100

Total number of persons = 300

P(E) = 100/300

The probability that someone prefers Frosted Flakes given that the person is an Adult is 1/4.

4) We have to find the probability that a person does not prefer Bran Flakes.

P(A person does not prefer Bran Flakes) = 1 - P(A person prefers Bran Flakes).

P(A person prefers Bran Flakes) = Number of persons who prefer Bran Flakes/total number of persons

Number of persons who prefer Bran Flakes = 80

Total number of persons = 300

Hence, P(A person prefers Bran Flakes) = 80/300

Hence, P(A person does not prefer Bran Flakes) = 1 - (80/300) = 220/300 = 11/15

The probability that a person does NOT prefer Bran Flakes is 11/15.