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Consider the following in respect of two events $\mathrm{A}$ and $\mathrm{B}$ :
$1.$ $\quad \mathrm{P}($ A occurs but not $\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$ if $\mathrm{B} \subset \mathrm{A}$
$2.$ $\quad \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone occurs $)=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$3.$ $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$ if $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive
Which of the above is/are correct?
Options:
$1.$ $\quad \mathrm{P}($ A occurs but not $\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$ if $\mathrm{B} \subset \mathrm{A}$
$2.$ $\quad \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone occurs $)=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$3.$ $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$ if $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive
Which of the above is/are correct?
Solution:
1474 Upvotes
Verified Answer
The correct answer is:
1 and 3 only
Statement $1:$ If $B \subset A$, then $\mathrm{P}(\mathrm{A}-\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$
$\therefore$ It is correct.
Statement $2: \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone $)$ $=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(-\mathrm{A} \cap \mathrm{B})$
$\therefore$ It is false. Statement $3:$ If $A, B$ are mutually exclusive events, then $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$
It is correct.
$=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$
$\therefore$ It is correct.
Statement $2: \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone $)$ $=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(-\mathrm{A} \cap \mathrm{B})$
$\therefore$ It is false. Statement $3:$ If $A, B$ are mutually exclusive events, then $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$
It is correct.
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