=+8. Let s∞ be the extinction probability of a supercritical branching process with progeny generating function Q(s) = ∞

k=0 qksk. If the mean

μ of Q(s) is fixed, then one can construct counterexamples showing that s∞ is not necessarily increasing as a function of q0. As a case in point, let Q(s) = P

1 2 + 1 2 s

and consider P(s) = 1 6 +

5 6

s3 P(s) = 3 32 +

15 24s2 +

5 32 s4 +

3 24s5.

Check numerically that these two choices lead to the extinction probabilities s∞ = 0.569 and s∞ = 0.594 and coefficients q0 = 0.271 and q0 = 0.264.