Question

If the events A and B are mutually exclusive, then $P\left( {}^{A}/{}_{B} \right)$ ?

Answer 1

In logic and probability theory, two events are mutually exclusive if they cannot both occur at the same time . An example is that of a dice. When we roll a dice, two numbers cannot appear at the same time. Only one out of the $6$ numbers can appear at a time. From this, we can infer that one event does not let the other event to happen. So we can conclude that the probability of their intersection happening is $0$.

Complete step by step solution:
These kind of events where happening of one event does not let the other event to happen are called the mutually exclusive event. When all these kind of events are put together, it makes up the sample space.
Sample space is nothing but the total expected outcome out of a situation.
So we already established that the probability of happening of intersection of two mutually exclusive events is $0$.
Let us write it in mathematical and logic terms
$\Rightarrow P\left( A\cap B \right)=0$ .
In the question, we are asked about conditional probability.
We know the basic formula behind conditional probability. It is the following :
$\Rightarrow P\left( {}^{A}/{}_{B} \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)}$ .
Let us use this formula to solve our question.
$\Rightarrow P\left( {}^{A}/{}_{B} \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)}$
Since , it is specified in the question that these two events are mutually exclusive , $P\left( A\cap B \right)=0$.
Let us substitute this and get the answer.
Upon substituting, we get the following :
$\begin{aligned} & \Rightarrow P\left( {}^{A}/{}_{B} \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)} \\\ & \Rightarrow P\left( {}^{A}/{}_{B} \right)=\dfrac{0}{P\left( B \right)} \\\ & \Rightarrow P\left( {}^{A}/{}_{B} \right)=0 \\\ \end{aligned}$

$\therefore$ If events A and B are mutually exclusive, then $P\left( {}^{A}/{}_{B} \right)=0$.

Note: It is very important to remember all the theorems in probability . We should be able to prove theorems such as the addition theorem, Bayes theorem. Problems from probability need a lot of practice. There is a lot of logic which is involved behind every problem. We should understand each and every step of the solution to be able to solve any kind of question from chapter. We should remember all the formulae and definitions as well.