Let [math]Z_1[/math], [math]Z_2[/math], ..., [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 eventually dies out if

• [math]p_0 = 1/2[/math], [math]p_1 = 1/4[/math], and [math]p_2 = 1/4[/math].
• [math]p_0 = 1/3[/math], [math]p_1 = 1/3[/math], and [math]p_2 = 1/3[/math].
• [math]p_0 = 1/3[/math], [math]p_1 = 0[/math], and [math]p_2 = 2/3[/math].
• [math]p_j = 1/2^{j + 1}[/math], for [math]j = 0[/math], 1, 2, ....
• [math]p_j = (1/3)(2/3)^j[/math], for [math]j = 0[/math], 1, 2, ....
• [math]p_j = e^{-2} 2^j/j![/math], for [math]j = 0[/math], 1, 2, ... (estimate [math]d[/math] numerically).