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Answer to Riddle #4 Mythical City Population of Boys
- A mythical city contains
100,000 married couples but no children. Each family wishes to “continue
the male line”, but they do not wish to over-populate. So, each family
has one baby per annum until the arrival of the first boy. For example,
if (at some future date) a family has five children, then it must be
either that they are all girls, and another child is planned, or that
there are four girls and one boy, and no more children are planned.
Assume that children are equally likely to be born male or female.
Let
p(t) be the percentage of children that are male at the end of year t.
How is this percentage expected to evolve through time?
I worked this little riddle out for myself also so
it's probably fairly easy...
Right unusually let me tell you the answer first:-
p(t) ≠ f(t)
p(t) = 50%
That is to say the percentage of children
that are male is not a function of time and is always 50%. This may seem a
little counter intuitive as we know that at some point in time a family could
conceivably have 10 girls and one boy, but this is balanced by the fact that
half the families will have no girls at all.
The table below is drawn up using the
simple rule that of the families who have a new girl one year all will try for
a new baby, with half of them having a boy and half having a girl.
| |
End of Year 1 |
Year 2 |
Year 3 |
Year 4 |
Year 5 |
Year
Infinity |
| Total Boys |
50,000 |
75,000 |
87,500 |
93,750 |
96,875 |
100,000 |
| Total Girls |
50,000 |
75,000 |
87,500 |
93,750 |
96,875 |
100,000 |
| New Boys |
50,000 |
25,000 |
12,500 |
6,250 |
3,125 |
0 |
| New Girls |
50,000 |
25,000 |
12,500 |
6,250 |
3,125 |
0 |
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