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\newtheorem*{example}{Example}
\newtheorem*{exercise}{Exercise}
\newtheorem*{conjecture}{Conjecture}
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\begin{document} 
\title{Written Assignment 3}
\section{Written Assignment 3}  
\subsection{Problem 1}
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On the red line, 25\% of inbound passengers arriving at the JFK/UMass station are from the Braintree branch, while 75\% are from the Mattapan branch.  22\% of the passengers from the Braintree branch leave the train at JFK/UMass, and 16\% of passengers from the Mattapan branch exit at JFK/UMass.
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a) What is the probability that a randomly chosen inbound passenger exits the train at JFK/UMass? 
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b) What is the probability that a randomly chosen inbound passenger who exits the train at JFK/UMass arrived on the Braintree branch? 
\subsection{Problem 2}
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A car manufacturer has three sources, $A$, $B$, and $C$ for fuel injectors.  $A$ supplies 45\% of the fuel injectors used, while $B$ supplies 35\% and $C$ supplies the remainder.  The fuel injectors are covered under a 50,000 mile waranty program, and warranty records indicate 3\% of injectors made by $A$, 2\% of those made by $B$, and 1\% of those made by $C$ fail in the first 50,000 miles.      
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a) What is the probability that a randomly chosen injector does not fail in the first 50,000 miles? 
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b) An injector is chosen at random from the injectors that failed in the first 50,000 miles. What is the probability that it was made by $A$? 
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c) An injector is chosen at random from the injectors that failed in the first 50,000 miles. What is the probability that it was made by $B$?
\subsection{Problem 3}
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Urn 1 contains one green, two blue, and three red chips.  Urn 2 contains two green, two red, and one blue chip.  An urn is chosen at random, with Urn 1 having a 40\% chance of being chosen, and a single chip is drawn from the chosen urn.
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a) What is the probability that a green chip is chosen? 
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b) What is the probability that a red chip is chosen? 
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c) If a green chip is chosen, what is the probability that it came from Urn 1? 
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d) If a red chip is chosen, what is the probability that it came from Urn 2?
\subsection{Problem 4}
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Urn 1 contains two blue, and four red chips.  Urn 2 contains three blue, and three red chips.  A coin is tossed and if the result is heads, a chip is drawn from Urn 1 and placed in Urn 2.  If the result is tails, a chip is drawn from Urn 2 and placed in Urn 1.  Next, a single chip is drawn from the urn that received the selected chip.   
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a) What is the probability that second chip drawn is red? 
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b) If the second draw results in a red chip, what is the probability that it was originally in Urn 1? 
\end{document}



