Editing Branching process (section)

== Example of extinction problem ==
Consider a parent can produce at most two offspring. The extinction probability in each generation is:

:<math>d_m=p_0+p_1d_{m-1}+p_2(d_{m-1})^2. \, </math>

with ''d''<sub>0</sub>&nbsp;=&nbsp;0. For the ultimate extinction probability, we need to find ''d'' which satisfies ''d''&nbsp;=&nbsp;''p''<sub>0</sub>&nbsp;+&nbsp;''p''<sub>1</sub>d&nbsp;+&nbsp;''p''<sub>2</sub>''d''<sup>2</sup>.

Taking as example probabilities for the numbers of offspring produced ''p''<sub>0</sub>&nbsp;=&nbsp;0.1, ''p''<sub>1</sub>&nbsp;=&nbsp;0.6, and ''p''<sub>2</sub>&nbsp;=&nbsp;0.3, the extinction probability for the first 20 generations is as follows:

{|class=wikitable
! Generation # (1–10) !! Extinction probability !! !! Generation # (11–20) !! Extinction probability
|-
| 1 || 0.1 || || 11 || 0.3156
|-
| 2 || 0.163 || || 12 || 0.3192
|-
| 3 || 0.2058 || || 13 || 0.3221
|-
| 4 || 0.2362 || || 14 || 0.3244
|-
| 5 || 0.2584 || || 15 || 0.3262
|-
| 6 || 0.2751 || || 16 || 0.3276
|-
| 7 || 0.2878 || || 17 || 0.3288
|-
| 8 || 0.2975 || || 18 || 0.3297
|-
| 9 || 0.3051 || || 19 || 0.3304
|-
| 10 || 0.3109 || || 20 || 0.331
|}

In this example, we can solve algebraically that ''d''&nbsp;=&nbsp;1/3, and this is the value to which the extinction probability converges with increasing generations.