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Two persons $\mathrm{P}$ and $\mathrm{Q}$ are considering to apply for a job. The probability that $\mathrm{P}$ applies for the job is $1 / 4$, the probability that $\mathrm{P}$ applies for the job given that $\mathrm{Q}$ applies for the job is $1 / 2$, and the probability that $Q$ applies for the job given that $P$ applies for the job is $1 / 3$. Then the probability that $\mathrm{P}$ does not apply for the job given that $\mathrm{Q}$ does not apply for the job is
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Verified Answer
The correct answer is:
$\frac{4}{5}$
Given $\mathrm{P}(\mathrm{P})=\frac{1}{4}, \mathrm{P}\left(\frac{\mathrm{P}}{\mathrm{Q}}\right)=\frac{1}{2}, \mathrm{P}\left(\frac{\mathrm{Q}}{\mathrm{P}}\right)=\frac{1}{3}$.
$$
\mathrm{P}\left(\frac{\overline{\mathrm{P}}}{\mathrm{Q}}\right)=?
$$
Now, $P\left(\frac{Q}{R}\right)=\frac{1}{3}$
$$
\begin{aligned}
& \frac{\mathrm{P}(\mathrm{P} \cap \mathrm{Q})}{\mathrm{P}(\mathrm{P})}=\frac{1}{3} \\
& \mathrm{P}(\mathrm{P} \cap \mathrm{Q})=\frac{1}{3} \times \frac{1}{4}=\frac{1}{12}
\end{aligned}
$$
So, $\frac{\mathrm{P}(\mathrm{P} \cap \mathrm{Q})}{\mathrm{P}(\mathrm{Q})}=\frac{1}{2}$
$$
\begin{aligned}
& P(Q)=2 \times \frac{1}{12}=\frac{1}{6} \\
& P(\bar{Q})=1-P(Q)=1-\frac{1}{6}=\frac{5}{6} \\
& P(\bar{P})=1-P(P)=1-\frac{1}{4}=\frac{3}{4} \\
& P(\bar{P} \cup \bar{Q})=1-P(P \cap Q) \\
& =1-\frac{1}{12}=\frac{11}{12}
\end{aligned}
$$
Then, $\mathrm{P}(\overline{\mathrm{P}} \cap \overline{\mathrm{Q}})=\mathrm{P}(\overline{\mathrm{P}})+\mathrm{P}(\overline{\mathrm{Q}})-\mathrm{P}(\overline{\mathrm{P}} \cup \overline{\mathrm{Q}})$
$$
\begin{aligned}
& =\frac{3}{4}+\frac{5}{6}-\frac{11}{12} \\
& =\frac{18+20-22}{24}=\frac{38-22}{24}=\frac{16}{24}=\frac{2}{3}
\end{aligned}
$$
Now, $\mathrm{P}\left(\frac{\overline{\mathrm{P}}}{\mathrm{Q}}\right)=\frac{\mathrm{P}(\overline{\mathrm{P}} \cap \overline{\mathrm{Q}})}{\mathrm{P}(\overline{\mathrm{Q}})}$
$$
=\frac{\frac{2}{3}}{\frac{5}{6}}=\frac{4}{5}
$$
$$
\mathrm{P}\left(\frac{\overline{\mathrm{P}}}{\mathrm{Q}}\right)=?
$$
Now, $P\left(\frac{Q}{R}\right)=\frac{1}{3}$
$$
\begin{aligned}
& \frac{\mathrm{P}(\mathrm{P} \cap \mathrm{Q})}{\mathrm{P}(\mathrm{P})}=\frac{1}{3} \\
& \mathrm{P}(\mathrm{P} \cap \mathrm{Q})=\frac{1}{3} \times \frac{1}{4}=\frac{1}{12}
\end{aligned}
$$
So, $\frac{\mathrm{P}(\mathrm{P} \cap \mathrm{Q})}{\mathrm{P}(\mathrm{Q})}=\frac{1}{2}$
$$
\begin{aligned}
& P(Q)=2 \times \frac{1}{12}=\frac{1}{6} \\
& P(\bar{Q})=1-P(Q)=1-\frac{1}{6}=\frac{5}{6} \\
& P(\bar{P})=1-P(P)=1-\frac{1}{4}=\frac{3}{4} \\
& P(\bar{P} \cup \bar{Q})=1-P(P \cap Q) \\
& =1-\frac{1}{12}=\frac{11}{12}
\end{aligned}
$$
Then, $\mathrm{P}(\overline{\mathrm{P}} \cap \overline{\mathrm{Q}})=\mathrm{P}(\overline{\mathrm{P}})+\mathrm{P}(\overline{\mathrm{Q}})-\mathrm{P}(\overline{\mathrm{P}} \cup \overline{\mathrm{Q}})$
$$
\begin{aligned}
& =\frac{3}{4}+\frac{5}{6}-\frac{11}{12} \\
& =\frac{18+20-22}{24}=\frac{38-22}{24}=\frac{16}{24}=\frac{2}{3}
\end{aligned}
$$
Now, $\mathrm{P}\left(\frac{\overline{\mathrm{P}}}{\mathrm{Q}}\right)=\frac{\mathrm{P}(\overline{\mathrm{P}} \cap \overline{\mathrm{Q}})}{\mathrm{P}(\overline{\mathrm{Q}})}$
$$
=\frac{\frac{2}{3}}{\frac{5}{6}}=\frac{4}{5}
$$
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