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Question:
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If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of $\mathrm{A}$. B occurs is q, then which of the following is/are correct?
$1.$ $\quad \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 \mathrm{p}-\mathrm{q}$
$2.$ $\quad \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{p}-\mathrm{q}$
Select the correct answer using the code given below:
Options:
$1.$ $\quad \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 \mathrm{p}-\mathrm{q}$
$2.$ $\quad \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{p}-\mathrm{q}$
Select the correct answer using the code given below:
Solution:
2009 Upvotes
Verified Answer
The correct answer is:
Both 1 and 2
$\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=q$
$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=p$
$\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=2 p+q$
$1-\mathrm{P}(\overline{\mathrm{A}})+1-\mathrm{P}(\overline{\mathrm{B}})=p+q$
$\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 p-q$
$\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})$
$=1-(q+p)=1-p-q$
$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=p$
$\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=2 p+q$
$1-\mathrm{P}(\overline{\mathrm{A}})+1-\mathrm{P}(\overline{\mathrm{B}})=p+q$
$\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 p-q$
$\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})$
$=1-(q+p)=1-p-q$
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