In a certain town, $60\%$ of the families own a car, $30\%$own a house and $20\%$ own both a car and a house. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both ?
- 0.5
- 0.7
- 0.1
- 0.9
The Correct Option is A
Solution and Explanation
be the set of families who own a house.
Then, $P(A)=60 \%, P(B)=30 \%$
and $P(A \cap B)=20 \%$
Now, $P(A \cup B) =P(A)+P(B)-P(A \cap B)$
$=60+30-20=70 \%$
Now, families who owns a car or a house but not both $=A \Delta B$
$\Rightarrow P(A \Delta B) =P(A \cup B)-P(A \cap B)$
$=70-20=50 \%=0.5$
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Concepts Used:
Bayes Theorem
Bayes’ Theorem is a part of the conditional probability that helps in finding the probability of an event, based on previous knowledge of conditions that might be related to that event.
Mathematically, Bayes’ Theorem is stated as:-
\(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\)
where,
- Events A and B are mutually exhaustive events.
- P(A) and P(B) are the probabilities of events A and B, respectively.
- P(A|B) is the conditional probability of the happening of event A, given that event B has happened.
- P(B|A) is the conditional probability of the happening of event B, given that event A has already happened.
This formula confines well as long as there are only two events. However, Bayes’ Theorem is not confined to two events. Hence, for more events.