Problem source

The twins Anna and Bella share a computer and never sign their e-mails.
 When I e-mail them, only the twin
currently online responds.  The
 probability that it is Anna who is online is $p$ and she answers each
 question I ask her truthfully with probability $a$, independently of all her
 other answers, even if a question is repeated. The probability that it is
 Bella  who is online is~$q$, where $q=1-p$, and she answers each question
 truthfully with probability $b$, independently of all her other answers,
 even if a question is repeated.
\begin{questionparts}
\item
I send the twins the  e-mail: 
`Toss a fair coin and answer the following  question. 
Did the coin come down heads?'. I receive the answer `yes'.
Show that the probability that the coin
did come down heads is $\frac{1}{2}$ if and
only if  $2(ap+bq)=1$.
\item
I send the twins the e-mail: 
`Toss a fair coin and answer  the following question. 
Did the coin come down heads?'. I receive the answer `yes'.
I then send the e-mail: `Did the coin come down heads?' and I receive
the answer `no'. Show that the probability (taking into
account these answers) that the coin did come down heads is $\frac{1}{2}\,$.
\item
I send the twins the e-mail: `Toss a fair coin and answer the following
question. Did the coin come down heads?'. I receive the answer `yes'.
I then send the e-mail: `Did the coin come down heads?' and I receive
the answer `yes'. Show that, if $2(ap+bq)=1$,
the probability (taking into account these answers) that the coin did
come down heads is $\frac{1}{2}\,$.
\end{questionparts}