Q1
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Probability
└── Homework
└── W4
└── Q1.tex
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\begin{document}
\section*{Question 1}
\noindent A student is applying to Harvard and Dartmouth. She estimates that she has a probability of $0.5$ of being accepted at Dartmouth and $0.3$ of being accepted at Harvard. She further estimates that the probability that she will be accepted by both is $0.2$.
\bigskip
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\bigskip\item What is the probability that she is accepted by Dartmouth if she is accepted by Harvard?
\subsection*{Solution}
Let $\it{accept}$ and $\it{reject}$ denote the results of an application, and let $D$ and $H$ denote the events of being accepted by Dartmouth and by Harvard, respectively. Define the probability space as
\[
\begin{aligned}
\Omega &= \{\mathrm{accept},\,\mathrm{reject}\}\\
\mathcal{F} &= \mathcal{P}(\Omega)\\
\bP &:\enspace \bP(D) = 0.5,\enspace \bP(H) = 0.3
\end{aligned}
\]
We know that
\[\bP(D\cap H) = 0.2\]
Then we have
\[\bP_H(D) = \frac{\bP(D\cap H)}{\bP(H)} = \frac{2}{3} \approx 0.667\]
\subsection*{Answer}
\[\boxed{\bP_H(D) = \frac{2}{3} \approx 0.667}\]
\bigskip\item Is the event “accepted at Harvard” independent of the event “accepted at Dartmouth”?
\subsection*{Solution}
Applying the Bayes’ Theorem, we know that
\[\bP_D(H) = \frac{\bP_H(D)\bP(H)}{\bP(D)} = 0.4 \ne 0.3 = \bP(H)\]
Then we know that $D$ and $H$ are not stochastically independent.
\subsection*{Answer}
\[\boxed{\text{No, $H$ is not stochastically independent of $D$.}}\]
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