Problem source

The random variable $X$ has mean $\mu$ and standard deviation $\sigma$. The distribution of $X$ is symmetrical about $\mu$ and satisfies:
\[\P \l X \le \mu + \sigma \r = a \mbox{  and   } \P \l X \le \mu + \tfrac{1}{ 2}\sigma \r = b\,,\]
 where $a$ and $b$ are fixed numbers. Do not assume that $X$ is Normally distributed.
\begin{questionparts}
\item  Determine expressions (in terms of $a$ and $b$) for 
\[ \P \l \mu-\tfrac12 \sigma \le X \le \mu + \sigma \r \mbox{  and  }  \P \l X \le \mu +\tfrac12 \sigma \; \vert \; X \ge \mu - \tfrac12 \sigma \r.\]
\item  My local supermarket sells cartons of skimmed milk and cartons of full-fat milk: $60\%$ of the cartons it sells contain skimmed milk, and the rest contain full-fat milk. 
The volume of skimmed milk in a carton is modelled by $X$ ml, with $\mu = 500$ and $\sigma =10\,$. The volume of full-fat milk in a carton is modelled by  $X$ ml, with $\mu = 495$ and $\sigma = 10\,$.
\begin{enumerate}
\item Today, I bought one carton of milk, chosen at random, from this supermarket.  When I get home, I find that it contains less than 505 ml.  Determine an expression (in terms of $a$ and $b$)  for the probability that this carton of milk contains more than 500 ml.
\item Over the years, I have bought a very large number of cartons of milk, all chosen at random, from this supermarket. $70\%$ of the cartons I have bought have contained at most 505 ml of milk.  Of all the cartons that have contained at least 495 ml of milk, one third of them have contained full-fat milk. Use this information to estimate the values of $a$ and $b$. 
\end{enumerate}
\end{questionparts}

Solution source

\begin{questionparts}
\item $\,$ \begin{align*}
&& \mathbb{P}\left (\mu - \tfrac12 \sigma \leq X \right) &=  \mathbb{P}\left (X \leq \mu + \tfrac12 \sigma \right) \tag{by symmetry} \\
&&&= b \\
\Rightarrow && \mathbb{P} \left (\mu - \tfrac12 \sigma \leq X \leq \mu + \sigma \right) &= a - (1-b) = a+b - 1\\
\\
&& \mathbb{P} \left ( X \le \mu +\tfrac12 \sigma \vert X \ge \mu - \tfrac12 \sigma \right ) &= \frac{ \mathbb{P} \left (\mu - \tfrac12 \sigma \leq X \leq \mu + \tfrac12 \sigma \right)}{\mathbb{P} \left ( X \ge \mu - \tfrac12 \sigma \right )} \\
&&&= \frac{b-(1-b)}{1-(1-b)} \\
&&&= \frac{2b-1}{b}
\end{align*}

\item \begin{enumerate}
\item Let $Y$ be the volume of milk in the carton I bring home, we are interested in:

\begin{align*}
&& \mathbb{P}(Y \geq 500 | Y \leq 505) &= \frac{\mathbb{P}(500 \leq Y \leq 505)}{\mathbb{P}(Y \leq 505)} \\
&&&=\frac{\mathbb{P}(500 \leq Y \leq 505|\text{skimmed})\mathbb{P}(\text{skimmed})+\mathbb{P}(500 \leq Y \leq 505|\text{full fat})\mathbb{P}(\text{full fat})}{\mathbb{P}(Y \leq 505|\text{skimmed})\mathbb{P}(\text{skimmed})+\mathbb{P}(Y \leq 505|\text{full fat})\mathbb{P}(\text{full fat})} \\
&&&= \frac{\frac35 \cdot \mathbb{P}(\mu \leq X \leq \mu + \tfrac12 \sigma) + \frac25 \cdot \mathbb{P}(\mu+\tfrac12 \sigma \leq X \leq \mu +\sigma)}{\frac35 \cdot \mathbb{P}(X \leq \mu + \tfrac12 \sigma) + \frac25 \cdot \mathbb{P}(X \leq \mu +\sigma)} \\
&&&= \frac{\frac35 \cdot(b-\tfrac12) + \frac25 \cdot (a-b)}{\frac35 \cdot b + \frac25 \cdot a} \\
&&&= \frac{b+2a-\frac32}{3b+2a} \\
&&&= \frac{4a+2b-3}{4a+6b}
\end{align*}

\item $70\%$ of cartons have contained at most 505 ml, so:

\begin{align*}
&& \tfrac7{10} &= \mathbb{P}(Y \leq 505) \\
&&&=  \mathbb{P}(Y \leq 505 | \text{ skimmed}) \mathbb{P}(\text{skimmed}) +  \mathbb{P}(Y \leq 505 | \text{ full fat}) \mathbb{P}(\text{full fat}) \\
&&&=  \mathbb{P}(X \leq \mu + \tfrac12 \sigma) \cdot \tfrac35 +  \mathbb{P}(X\leq \mu + \sigma ) \cdot \tfrac25 \\
\Rightarrow && 7 &= 6b+ 4a
\end{align*}

$\tfrac13$ of cartons containing 495 ml contained full fat milk:

\begin{align*}
&& \tfrac13 &= \mathbb{P}(\text{full fat} | Y \geq 495) \\
&&&= \frac{\mathbb{P}(\text{full fat and}  Y \geq 495) }{\mathbb{P}(Y \geq 495)} \\
&&&= \frac{\mathbb{P}(X \geq \mu)\frac25}{\mathbb{P}(X \geq \mu)\cdot \frac25+\mathbb{P}(X \geq \mu-\tfrac12 \sigma)\cdot \frac35} \\
&&&= \frac{\frac15}{\frac12 \cdot \frac25 + b\frac35}\\
&&&= \frac{1}{1+ 3b }\\
\Rightarrow && 3b+1 &= 3 \\
\Rightarrow && b &= \frac23 \\
&& a &= \frac34
\end{align*}

\end{enumerate}
\end{questionparts}