Suppose that you are interested in the proportion of females in a certain organization and that as a first step in your investigation you intend to find out the sex of the first 11 members on the membership list. Before doing so, you have prior beliefs which you regard as equivalent to 25% of this data, and your prior beliefs suggest that a third of the membership is female.

Suggest a suitable prior distribution and find its standard deviation. 

Suppose that 3 of the first 11 members turn out to be female; find your posterior distribution and give a 50% posterior HDR for this distribution.

Find the mean, median and mode of the posterior distribution.

Would it surprise you to learn that in fact 86 of the total number of 433 members are female?