Question 1029372: A cereal company has been running a radio advertisement for a new cereal. A marketing research firm determined that probability that an individual has heard the advertisement is 0.5, the probability that an individual bought the cereal is 0.2 , and the probability that an individual has heard the advertisement and bought the cereal is 0.1.
a) What is the probability that an individual will purchase the product, given that the person has heard the advertisement?
b) What is the probability that an individual has heard the advertisement, given that the person purchased the product?
c) Are the two events, the person purchased the product, and an individual has heard the advertisement independent events? Justify your answer.
Answer by mathmate(429)   (Show Source):
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Question:A cereal company has been running a radio advertisement for a new cereal. A marketing research firm determined that probability that an individual has heard the advertisement is 0.5, the probability that an individual bought the cereal is 0.2 , and the probability that an individual has heard the advertisement and bought the cereal is 0.1.
a) What is the probability that an individual will purchase the product, given that the person has heard the advertisement?
b) What is the probability that an individual has heard the advertisement, given that the person purchased the product?
c) Are the two events, the person purchased the product, and an individual has heard the advertisement independent events? Justify your answer.
Solution:
First we recognize that the problem involves two events,
H = an individual has heard the advertisement of the cereal
B = an individual has bought the cereal
Then we need to interpret the given information in terms of the two events.
P(H)=0.5
P(B)=0.2
and, most importantly,
P(B∩H)=0.1, meaning the individual bought the cereal AND he bought the product, any one event can precede the other.
a) What is the probability that an individual will purchase the product, given that the person has heard the advertisement?
Recall that conditional probability is defined as
P(A|B)=Probability of event A happening given that B has already occurred
=P(A∩B)/P(B)
Here we already know P(H∩B)=P(B∩H) and P(B), so we only need to substitute:
P(B|H)=P(B∩H)/P(H)=P(H∩B)/P(H)=0.1/0.5=0.2
b)What is the probability that an individual has heard the advertisement, given that the person purchased the product?
For this we are looking for
P(H|B)=P(H∩B)/P(B)=0.1/0.2=0.5
c)Are the two events, the person purchased the product, and an individual has heard the advertisement independent events? Justify your answer.
Two events A and B are independent if and only if P(A)*P(B)=P(A∩B).
So we check
P(H)*P(B)=0.5*0.2=0.1 which equals P(H∩B), therefore the two events are independent.
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