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\lhead{\textbf{Lizzy Hanley}\\\textbf{ Molly Soja}
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\rhead{Solutions to extra credit: Lecture 22}

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\begin{problem}{}
\textbf{Q1:} 6 cards are drawn from a deck of 52 cards.  Let X be the number of kings that are drawn.  What is the distribution (and the corresponding parameters) of X?\\
\textbf{Solution}:
\textbf{}Hypergeometric Distribution because we are trying to find the probability of getting a certain amount of kings.\\
\textbf{} m = 140 r = 4 n = 6
$$ P(X = k) = \frac{\binom{4}{X} \binom{48}{6-X}}{\binom{52}{6}}$$


\end{problem}




\begin{problem}{}
\textbf{Q2:} You have 40 red socks, 20 yellow socks, and 80 blue socks that are in a bag.  This morning, you do the following.  You randomly take out a pair of socks, if the colors do not match, you put them back in and repeat.  Let X be the number of trials until you get a pair with matched color.  What is the distribution (and the corresponding parameters) of X?\\
\textbf{Solution}:
\textbf{}Negative Binomial Distribution because we want the probability of getting one pair in n tries. \\
\textbf{}k = 1
$$P(X = n) = \binom{n-1}{0}\bigg(\frac{\binom{40}{2}}{\binom{140}{2}} + \frac{\binom{10}{2}}{\binom{140}{2}} + \frac{\binom{20}{2}}{\binom{140}{2}}\bigg)^1\bigg(1-\bigg(\frac{\binom{40}{2}}{\binom{140}{2}} + \frac{\binom{10}{2}}{\binom{140}{2}} + \frac{\binom{20}{2}}{\binom{140}{2}}\bigg)\bigg)^{n-1}$$




\end{problem}




\begin{problem}{}
\textbf{Q3:} In expectation, the number of rainy days per year in San Diego is 60.  Let X be the number of rainy days next year.  What is the distribution(and the corresponding parameters) of X?\\
\textbf{Solution}:
\textbf{}Poisson Distribution because we want the probability of seeing the event given what we expect will happen.\\
$$\lambda = 60 $$
$$P(X = k) = \frac{e^{-60}60^k}{k!}$$


\end{problem}
\newpage\pagebreak
\begin{problem}{}
\textbf{Q4:} You  have  infinitely  many  socks  in  a  bag.   This  morning,  you prefer  to  wear  red  socks  so  you  do  the  following.   You  randomly  take  out  a single sock until you have a pair of red socks.  The probabilities that you get a  specific  color  among{red,  yellow,  blue}are  equal.   Let X be  the  number of  trials  until  you  get  a  pair  of  red  socks.   What  is  the  distribution  (and  the corresponding parameters) of X?\\
\textbf{Solution}:
\textbf{}Negative Binomial Distribution because we want the probability of getting 2 same colored socks to get a pair in n tries. \\
\textbf{}k = 2
$$P(X = n) = \binom{n}{1}p^2(1-p)^{n-2}$$


\end{problem}





\begin{problem}{}
\textbf{Q5:} There are 6 bulbs in a house out of which 3 are defective.  If 2 bulbs are picked randomly, find the probability that at least one is defective.\\
\textbf{Solution}:
\textbf{} Hypergeometric Distribution because we want to know the probability of getting a certain amount of defective bulbs\\k = 1
$$P(X = k) = \binom{6}{1}\bigg(\frac{\binom{3}{1}\binom{6-3}{2-1}}{\binom{6}{2}} + \frac{\binom{3}{2}\binom{6-3}{2-2}}{\binom{6}{2}}\bigg)$$



\end{problem}

\textbf{Solve the following problems using the distributions that we discussed.}

\begin{problem}{}
\textbf{Q6:} John is taking part in four competitions.  If the probability of him winning any competition is 0.3, find the probability of him winning at least one competition.\\
\textbf{Solution}:
\textbf{}Binomial Distribution because


\end{problem}


\begin{problem}{}
\textbf{Q7:} The probability that a kid getting an A for a paper is 0.05, find the probability of at most 2 out of 10 kids getting A grade in that paper.\\
\textbf{Solution}:



\end{problem}







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