Question

If A and B are two mutually exclusive and exhaustive events with $P(B) = 3P(A)$, then what is the value of $P(\overline B )$?

Answer 1

Hint: In order to solve this question we need to first know about the terms Mutually Exclusive and Exhaustive Events . The two events are mutually exclusive if they cannot both be true . A clear example is the set of outcomes of a single coin toss , which can result in their head or tail , but not both . And Mutually exhaustive events are a set of events where at least one of the events must occur . Keeping these two terms in mind we are going to solve the question . Also the property of probability of happening of an event and the probability of not happening of an event is equal to 1

Complete step by step answer:
The prerequisite of the above question is to know how to apply the concepts of Mutually Exclusive and Exhaustive Events by just applying their properties which are as follows
Probability of intersection of event A and event B = $P(A \cap B) = 0$. Also,
Probability of union of event A and event B = $P(A \cup B) = 1$.
Now to apply the above identities or we can say properties we will put one formula which will make use of both of the above properties $P(A \cap B) = 0$ and $P(A \cup B) = 1$ that is
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Now , applying the properties given as per the question , we get
$
  P(A \cup B) = P(A) + P(B) - P(A \cap B) \\
  1 = P(A) + 3P(A) - 0 \\
  1 = 4P(A) - 0 \\
  \dfrac{1}{4} = P(A) \\
 $
 $P(B) = 3P(A) = 3 \times \dfrac{1}{4} = \dfrac{3}{4}$.
According to the question , we need to find the probability of complement B , $P(\overline B )$ .
We know the property of probability of happening of an event and the probability of not happening of an event is equal to 1 .
\[\]$
  P(B) + P(\overline B ) = 1 \\
  P(\overline B ) = 1 - P(B) \\
  P(\overline B ) = 1 - \dfrac{3}{4} \\
  P(\overline B ) = \dfrac{1}{4} \\
 $
the required $P(\overline B ) = \dfrac{1}{4}$.

Note: The likelihood of an event to occur or the extent to which the event is probable is called probability .
 The basic rule of probability used in the above question are :
The addition rule will apply when there is a union of 2 other events or we have to find one term when all 3 terms are given : $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
The complementary rule will apply whenever an event is a complement of another event that is $P(\overline B ) = 1 - P(B)$.
There are many more formulae which is used in different probability questions which arise due to different situations and events .
Ensure that you make the correct selection of formula for solving the probability questions .
Always remember that the likelihood of all events in a sample space adds up to 1 .
To check if your probability answer is correct or not , always remember that the likelihood of an event ranges from 0 to 1 , where 0 means the event to be an impossible one , and one represents the event to be certain .