In a Galton-Watson branching process starting with a single individual in generation zero, the offspring distribution is binomial B(3, 0.5). Find the probability that the process becomes extinct at the third generation. Provide your answer in four decimal places. Solution The Galton–Watson process is a branching stochastic process arising from Francis Galton\'s statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). While this is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups – and is also of use in understanding other processes (as described below) – its application to actual extinction of family names is fraught. In practice family names change for many other reasons, and dying out of name line is only one factor, as discussed in examples, below; the Galton–Watson process is thus of limited applicability in understanding actual family name distributions. In the classical Galton–Watson process described above, only men are considered, effectively modeling reproduction as asexual. A model more closely following actual sexual reproduction is the so-called \'bisexual Galton–Watson process\', where only couples reproduce. (Bisexual in this context refers to the number of sexes involved, not sexual orientation.) In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called \'mating function\' determines how many couples will form in a given generation. As before, reproduction of different couples are considered to be independent of each other. Now the analogue of the trivial case corresponds to the case of each male and female reproducing in exactly one couple, having one male and one female descendent, and that the mating function takes the value of the minimum of the number of males and females (which are then the same from the next generation onwards). Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as it is the case in the classical Galton–Watson process. However, excluding the non-trivial case, the concept of the averaged reproduction mean (Bruss (1984)) allows for a general sufficient condition for final extinction, treated in the next section. Citing historical examples of Galton–Watson process is complicated due to the history of family names often deviating significantly from the theoretical model. Notably, new names can be created, existing names can be changed over a person\'s lifetime, and people historically have often assumed names of unrelated persons, particularly nobility. Thus, a .

1. In a Galton-Watson branching process starting with a single individual in generation zero, the
offspring distribution is binomial B(3, 0.5). Find the probability that the process becomes extinct
at the third generation. Provide your answer in four decimal places.
Solution
The Galton–Watson process is a branching stochastic process arising from Francis
Galton's statistical investigation of the extinction of family names. The process models family
names as patrilineal (passed from father to son), while offspring are randomly either male or
female, and names become extinct if the family name line dies out (holders of the family name
die without male descendants). While this is an accurate description of Y chromosome
transmission in genetics, and the model is thus useful for understanding human Y-chromosome
DNA haplogroups – and is also of use in understanding other processes (as described below) –
its application to actual extinction of family names is fraught. In practice family names change
for many other reasons, and dying out of name line is only one factor, as discussed in examples,
below; the Galton–Watson process is thus of limited applicability in understanding actual family
name distributions. In the classical Galton–Watson process described above, only men are
considered, effectively modeling reproduction as asexual. A model more closely following actual
sexual reproduction is the so-called 'bisexual Galton–Watson process', where only couples
reproduce. (Bisexual in this context refers to the number of sexes involved, not sexual
orientation.) In this process, each child is supposed as male or female, independently of each
other, with a specified probability, and a so-called 'mating function' determines how many
couples will form in a given generation. As before, reproduction of different couples are
considered to be independent of each other. Now the analogue of the trivial case corresponds to
the case of each male and female reproducing in exactly one couple, having one male and one
female descendent, and that the mating function takes the value of the minimum of the number
of males and females (which are then the same from the next generation onwards). Since the
total reproduction within a generation depends now strongly on the mating function, there exists
in general no simple necessary and sufficient condition for final extinction as it is the case in the
classical Galton–Watson process. However, excluding the non-trivial case, the concept of the
averaged reproduction mean (Bruss (1984)) allows for a general sufficient condition for final
extinction, treated in the next section. Citing historical examples of Galton–Watson process is
complicated due to the history of family names often deviating significantly from the theoretical
model. Notably, new names can be created, existing names can be changed over a person's
lifetime, and people historically have often assumed names of unrelated persons, particularly
nobility. Thus, a small number of family names at present is not in itself evidence for names
having become extinct over time, or that they did so due to dying out of family name lines – that

2. requires that there were more names in the past and that they die out due to the line dying out,
rather than the name changing for other reasons, such as vassals assuming the name of their lord.
Chinese names are a well-studied example of surname extinction: there are currently only about
3,100 surnames in use in China, compared with close to 12,000 recorded in the past,[1][2] with
22% of the population sharing three family names (numbering close to 300 million people), and
the top 200 names covering 96% of the population. Names have changed or become extinct for
various reasons such as people taking the names of their rulers, orthographic simplifications,
taboos against using characters from an emperor's name, among others.[2] While family name
lines dying out may be a factor in the surname extinction, it is by no means the only or even a
significant factor. Indeed, the most significant factor affecting the surname frequency is other
ethnic groups identifying as Han and adopting Han names.[2] Further, while new names have
arisen for various reasons, this has been outweighed by old names disappearing.[2] By contrast,
some nations have adopted family names only recently. This means both that they have not
experienced surname extinction for an extended period, and that the names were adopted when
the nation had a relatively large population, rather than the smaller populations of ancient
times.[2] Further, these names have often been chosen creatively and are very diverse. Examples
include: Japanese names, which in general use date only to the Meiji restoration in the late 19th
century (when the population was over 30,000,000), have over 100,000 family names, surnames
are very varied, and the government restricts married couples to using the same surname. Many
Dutch names have only included a family name since the Napoleonic Wars in the early 19th
century, and there are over 68,000 Dutch family names. Thai names have only included a family
name since 1920, and only a single family can use a given family name, hence there are a great
number of Thai names. Further, Thai people change their family names with some frequency,
complicating the analysis.