[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A coin has an unknown bias [math]p[/math] that is assumed to be uniformly

distributed between 0 and 1. The coin is tossed [math]n[/math] times and heads turns up [math]j[/math] times and tails turns up [math]k[/math] times. We have seen that the probability that heads turns up next time is

[[math]] \frac {j + 1}{n + 2}\ . [[/math]]

Show that this is the same as the probability that the next ball is black for the Polya urn model of Exercise \ref{sec 4.1}.. Use this result to explain why, in the Polya urn model, the proportion of black balls does not tend to 0 or 1 as one might expect but rather to a uniform distribution on the interval [math][0,1][/math].