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{\bf Question}

In a branching chain the number of offspring of any individual has
a binomial distribution with $n = 3,\ p = \frac{1}{2}.$  Find the
probability $P$ of extinction.



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{\bf Answer}

$Z_i \sim B(3, \frac{1}{2})$

So the p.g.f. is $\ds A(s) = \frac{1}{8} + \frac{3}{8}s +
\frac{3}{8}s^2 + \frac{1}{8}s^3$

So we have to solve $s^3 + 3s^2 - 5s + 1 = 0$

i.e. $(s-1)(s^2 + 4s - 1) = 0$

$s = 1 ,  \, \, -2 \pm \sqrt 5$

So $P = \sqrt 5 - 2 = 0.236\ldots$


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