Question

If $A$ and $B$ are any two events associated with random experiment, then by the definition of probability, which of following is true?
(i) $P(A\cup B)=\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in A\cup B$
(ii) $P(A)+P(B)=P(A\cup B)+P(A\cap B)$

• A. Only (i)
• B. Only (ii)
• C. Both (i) and (ii)
• D. None of these

Answer 1

Answer: (C)

In general, if $A$ and $B$ are any two events associated with a random experiment, then by the definition of probability of an event, we have
$P(A\cup B)=\Sigma P\left({\omega }_i\right),\forall {\omega }_i\in A\cup B$
Since, $A\cup B=(A-B)\cup (A\cap B)\cup (B-A)$,
We have
$P(A\cup B)$
$=\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in (A-B)\right]+\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in A\cap B\right]$
$+\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in B-A\right]$
(because $A-B,A\cap B$ and $B-A$ are mutually exclusive) .....(i)
Also, $P(A)+P(B)=\left[\Sigma p\left({\omega }_i\right)\forall {\omega }_i\in A\right]+\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in B\right]$
$=\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in (A-B)\cup (A\cap B)\right]+$
$\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in (B-A)\cup (A\cap B)\right]$
$=\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in (A-B)\right]+\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in (A\cap B)\right]+$
$\left[\Sigma P\left({\omega }_t\right)\forall {\omega }_l\in (B-A)\right]+\left[\Sigma P\left({\omega }_l\right)\forall {\omega }_l\in (A\cap B)\right]$
$=P(A\cup B)+\left[\Sigma P\left({\omega }_i\right)\forall {\omega }_i\in A\cap B\right]$ [using Eq. (i) $]$
$=P(A\cup B)+P(A\cap B)$
Hence, $P(A)+P(B)=P(A\cup B)+P(A\cap B)$