A coin is either fair or has two heads. You initially assign probabilities of 0.5 to each…

A coin is either fair or has two heads. You initially assign
probabilities of 0.5 to each possibility. The coin is then tossed twice, with
two heads appearing. Use Bayes’ theorem to work out the posterior probabilities
of each possible outcome.

The average income of a group of people is £8000. 80% of the
group have incomes within the range £6000–10 000. What is the minimum value of
the standard deviation of the distribution?