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.
- 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 \(2(ap+bq)=1\).
- 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 \(\frac{1}{2}\,\).
- 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 \(\frac{1}{2}\,\).