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Question: If \(A\) and \(B\) are two independent events , then \(P\left( {}^{A}/{}_{B} \right)\) ?...

If AA and BB are two independent events , then P(A/B)P\left( {}^{A}/{}_{B} \right) ?

Explanation

Solution

In logic and probability theory, two events are independent if the happening of one event does not affect the other event . An example is that of flipping a coin. When we first flip a coin, we may first get a head. And when we flip the coin, we may get a tail. One event is not stopping the other event or hindering the other event. So we can conclude that two events are independent if the occurrence or non – occurrence of one event does not affect the other one.

Complete step by step solution:
Since the occurrence or non – occurrence of an event does not affect the other one, we can safely arrive at a conclusion with respect to their probabilities.
The conclusion is :
⇒P(A∩B)=P(A)P(B)\Rightarrow P\left( A\cap B \right)=P\left( A \right)P\left( B \right)
In the question, we are asked about conditional probability.
Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred.
We know the basic formula behind conditional probability. It is the following :
⇒P(A/B)=P(A∩B)P(B)\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.
⇒P(A/B)=P(A∩B)P(B)\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 independent , P(A∩B)=P(A)P(B)P\left( A\cap B \right)=P\left( A \right)P\left( B \right).
Let us substitute this and get the answer.
Upon substituting, we get the following :
⇒P(A/B)=P(A∩B)P(B) ⇒P(A/B)=P(A)P(B)P(B) ⇒P(A/B)=P(A) \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{P\left( A \right)P\left( B \right)}{P\left( B \right)} \\\ & \Rightarrow P\left( {}^{A}/{}_{B} \right)=P\left( A \right) \\\ \end{aligned}

∴\therefore If events AA and BBare independent, then P(A/B)=P(A)P\left( {}^{A}/{}_{B} \right)=P\left( A \right).

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.