[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]

In the problem of points, discussed in

the historical remarks in Section \ref{sec 3.2}, two players, A and B, play a series of points in a game with player A winning each point with probability [math]p[/math] and player B winning each point with probability [math]q = 1 - p[/math]. The first player to win [math]N[/math] points wins the game. Assume that [math]N = 3[/math]. Let [math]X[/math] be a random variable that has the value 1 if player A wins the series and 0 otherwise. Let [math]Y[/math] be a random variable with value the number of points played in a game. Find the distribution of [math]X[/math] and [math]Y[/math] when [math]p = 1/2[/math]. Are [math]X[/math] and [math]Y[/math] independent in this case? Answer the same questions for the case [math]p = 2/3[/math].