Question 1164160: In a pet store, a customer is randomly selected. It was determined that the probability that they owned a dog was 60% and the probability they owned a cat was 50%. The probability that the customer owner both a cat and a dog was 23%. Are the events “owning a cat” and “owning a dog” independent or dependent events? Justify answer.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52835)   (Show Source):
You can put this solution on YOUR website! .
According to the definition of independent events, we should check if
P(owning a dog AND owning the cat) = P(owning a dog)*P(owning a cat).
The left side is 0.23 (given).
The right side is 0.6*0.5 = 0.3.
The two sides values are different, so the events ARE NOT INDEPENDENT. ANSWER
Solved, answered and explained // (justified).
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website! In a pet store, a customer is randomly selected. It was determined that the
probability that they owned a dog was 60% and the probability they owned a
cat was 50%. The probability that the customer owner both a cat and a dog
was 23%. Are the events “owning a cat” and “owning a dog” independent or
dependent events? Justify answer.
Two events are independent if the probability of either one is not changed
when we know the probability of the other. That is, they will be
independent if
P(D|C) = P(D) and P(C|D) = P(C)
P(D and C) 0.23
P(D|C) = —————————— = ———— = 0.46
P(C) 0.50
But P(D) = 0.60
So the probability of owning a dog is reduced if it is known that they own
a cat. So they are not independent.
We might as well investigate P(C|D)
P(C|D) = P(C)
P(C and D) 0.23
P(C|D) = —————————— = ———— = 0.3833...
P(D) 0.60
But P(C) = 0.50
So also the probability of owning a cat is reduced if it is known that they
own a dog. So they are not independent. So they are dependent.
Such dependent events are sometimes called "mutually restrictive", which
means that the probability is less when the other is given.
Events are sometimes called "mutually supportive" if the probability is
greater when the other is given.
There is a shortcut which is too cookbook for me, even though it is given in
every textbook, because it's quick and it works. I don't like it because it
doesn't teach you anything. That way is:
Two events are independent if the probability that they own both equals the
product of the probabilities that they own each:
So P(D)•P(C) = (0.60)•(0.50) = 0.30 but P(D and C) = 0.23. Those are not
equal so the events are dependent.
But it doesn't tell us what it really means for two events to be mutually
exclusive, merely restrictive, mutually independent, mutually supportive or
mutually guaranteeing. It's just a cookbook rule that works.
Edwin
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