\problemdata     
{S01-PS9-3}           
{probability-intro}              
{} 		
{S98-PS9, S01-PS9}
{S01 Karger}



\begin{problem}



There are 4 coins in a box, namely $c_1, c_2, c_3, c_4$. The probability
$p_i$ that the coin $c_i$ gives a Head is $\frac{1}{i}$ for $1\leq i\leq4$.
A coin is selected from the box randomly, and is being tossed until a Head is
seen. 

\bparts
\ppart
Write down the sample space of the experiment
\solution{
The sample space is
$\{(c_1,H)\}\cup\{c_2,c_3,c_4\}\times\{H,TH,TTH,TTTH,\dots\}$
}
\ppart
What is the probability that a Head is first seen in the $2^{nd}$ toss? 
\solution{
For this part and the following part, let $H_i$ be the event that ``a Head is first seen in the $i^{th}$ toss''. Let $C_i$ be the event that ``$c_i$ is drawn from the box''.

\begin{eqnarray*}
P(H_2)  &=&P(H_2|C_1)P(C_1)+P(H_2|C_2)P(C_2)+P(H_2|C_3)P(C_3)+P(H_2|C_4)P(C_4)\\
        &=&0\cdot\frac{1}{4}+\underbrace{\frac{1}{2}\cdot\frac{1}{2}}_{P(H_2|C_2)}\cdot\frac{1}{4}+\underbrace{\frac{2}{3}\cdot\frac{1}{3}}_{P(H_2|C_3)}\cdot\frac{1}{4}+\underbrace{\frac{3}{4}\cdot\frac{1}{4}}_{P(H_2|C_4)}\cdot\frac{1}{4}\\
        &=&\frac{95}{576}\\
\end{eqnarray*}
}
\ppart
Given that a Head is first seen in the $2^{nd}$ toss, what is the 
probability that $c_i$ is selected from the box?
\solution{
\begin{eqnarray*}
P(C_i|H_2)&=&\frac{P(C_i\cap H_2)}{P(H_2)}\\
         &=&\frac{P(H_2|C_i)P(C_i)}{P(H_2)}\\
         &=&\frac{\frac{i-1}{i}\cdot\frac{1}{i}\cdot\frac{1}{4}}{\frac{95}{576}}\\
         &=&\frac{144(i-1)}{95i^2}
\end{eqnarray*}

{\bf Checking:} $P(C_1|H_2)+P(C_2|H_2)+P(C_3|H_2)+P(C_4|H_2)=0+\frac{144}{95\cdot4}+\frac{144\cdot2}{95\cdot9}+\frac{144\cdot3}{95\cdot16}=1$. This makes sense.
}
\eparts

\end{problem}
