Probability for Ben, Amos and Carl Three men Amos, Ben and Carl share an office at work with a single telephone. Calls call in at random with the proportions of $\dfrac{1}{2}$ for Amos, $\dfrac{1}{3}$ for Ben and $\dfrac{1}{6}$ for Carl. For any incoming, calls, the probabilities that it will be picked up by Amos, Ben, and Carl are $\dfrac{1}{2}$, $\dfrac{3}{10}$ and $\dfrac{1}{5}$ respectively. For calls arriving during working hours, find the probability that (i)  a call is not picked up by the person being called, (ii)
a call is for Ben given that a call is not picked up by the person being called.
For (i), I tried to get P(proportion for Amos & not picked up) or P(proportion for Ben & not picked up) or P(proportion for Carl & not picked up)
$= \dfrac{1}{2} * \dfrac{1}{2} + \dfrac{1}{3} * \dfrac{1}{2} + \dfrac{1}{6} * \dfrac{5}{6} = 0.5546 $
But the answer for (i) is $\dfrac{37}{60}$, what went wrong?
How do I do for (ii)?
Your help is appreciated. Thanks
 A: You got one number wrong. Your idea is correct, we have
\begin{align*}
  \def\P{\mathbf P}\P(\text{call picked up by wrong person}) 
   &=\underbrace{ \frac 12}_{\text{call for A}}\cdot \underbrace{\frac 12}_{\text{call not picked by A}} +
  \underbrace{ \frac 13}_{\text{call for B}}\cdot \underbrace{\frac 7{10}}_{\text{call not picked by B}}+ 
\underbrace{ \frac 16}_{\text{call for C}}\cdot \underbrace{\frac 45}_{\text{call not picked by C}}\\
  &= \frac 14 + \frac 7{30} + \frac 4{30}\\
  &= \frac{30 + 28 + 16}{120}\\
  &= \frac{37}{60}
\end{align*}
For (ii), we have
\begin{align*}
  \P(\text{call for B}\mid\text{wrong person}) &= \frac{\P(\text{call for B}\cap \text{wrong person})}{\P(\text{wrong person})}\\
&= \frac{\frac 12 \cdot \frac 12}{\frac{37}{60}}\\
 &= \frac{15}{37}
\end{align*}
A: Part (i) should be $\frac{3\cdot5+2\cdot7+1\cdot8}{60}=\frac{37}{60}$.
Part (ii) is the probability that Amos or Carl picked up the phone, times the probability that the phone call was for Ben, divided by the probability that the phone call was picked up by the wrong person.
$$\frac{\frac{1}{3}(\frac12+\frac15)}{\frac{37}{60}}$$
$$=\frac{\frac{7}{30}}{\frac{37}{60}}$$
$$=\frac{14}{37}$$
