Girls and Boys
In a village, every child is born a girl with probability \(0.5\) and is born a boy with probability
\(0.5\). In this village, every family
keeps giving birth to children until the last child is a girl. What will
be the ratio of the population of girls and boys in this village?
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1674023738541498368
The ratio of girls and boys will be equal.
With this strategy, the number of girls in each family will be exactly
1. The number of boys is equal to:
zero with a probability of 1/2,
one with a probability of 1/4,
two with a probability of 1/8,
and in general \(i\) with probability \(2^{-i-1}\).
Let us denote this value by \(p\).
Therefore we have \(p = \sum_{i=0}^{\infty}
2^{-i-1}\) and thus \[2p -
(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...) = p.\] Notice
that the value inside the parenthesis is equal to 1, and therefore the
value of \(p\) will also be equal to
1.
Link to the solution on Twitter: https://twitter.com/Riazi_Cafe/status/1674756017924579328