Question

\[P\left( A\bigcup B \right)=P\left( A\bigcap B \right)\], if and only if the relation between \[P\left( A \right)\] and \[P\left( B \right)\] is \[P\left( A \right)+P\left( B \right)=2P\left( AB \right)\]. If this is true enter 1, else enter 0.

Answer 1

Hint:We will be using the concept of probability addition to tackle this question. Probability addition means that the probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur. So we will be using this formula \[P\left( A\bigcup B \right)=P\left( A \right)+P\left( B \right)-P(A\bigcap B)\] to prove the given expression.

Complete step-by-step answer:
Before proceeding with this question, we should know some definitions in probability.
Two events are mutually exclusive or disjoint if they cannot occur at the same time.
The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P (A ∩ B). If Events A and B are mutually exclusive, P (A ∩ B) = 0.
If the occurrence of Event A changes the probability of Event B, then Events A and B are dependent. On the other hand, if the occurrence of Event A does not change the probability of Event B, then Events A and B are independent.
The probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur.
 \[\Rightarrow P\left( A\bigcup B \right)=P\left( A \right)+P\left( B \right)-P(A\bigcap B).......(1)\]
It is mentioned in the question that \[P\left( A\bigcup B \right)=P\left( A\bigcap B \right).......(2)\]. So using equation (2) in equation (1) we get,
 \[\Rightarrow P\left( A\bigcap B \right)=P\left( A \right)+P\left( B \right)-P(A\bigcap B).......(2)\]
Rearranging equation (2) we get,
 \[\Rightarrow P\left( A \right)+P\left( B \right)=2P(A\bigcap B).......(3)\]
Notation: \[A\bigcap B=AB\]. So applying this in equation (3) we get,
 \[\Rightarrow P\left( A \right)+P\left( B \right)=2P(AB)\]
This is true so enter 1.

Note: Students should be thorough with the basic definitions and formulas of probability.They should remember the important formula of probability of addition i.e \[P\left( A\bigcup B \right)=P\left( A \right)+P\left( B \right)-P(A\bigcap B)\] for solving these types of questions.