[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]

Let [math]Z_1[/math], [math]Z_2[/math], \ldots, [math]Z_N[/math] describe a branching process in

which each parent has [math]j[/math] offspring with probability [math]p_j[/math]. Find the probability [math]d[/math] that the process dies out if

• [math]p_0 = 1/2[/math], [math]p_1 = p_2 = 0[/math], and [math]p_3 = 1/2[/math].
• [math]p_0 = p_1 = p_2 = p_3 = 1/4[/math].
• [math]p_0 = t[/math], [math]p_1 = 1 - 2t[/math], [math]p_2 = 0[/math], and [math]p_3 = t[/math], where [math]t \leq 1/2[/math].