Hello

I'm looking for a mathematical formula, that will give the average depth of a of a split recursion process,
where the chance of "going deeper" is constant.
(I know this is more f a mathematical than a perl question, but I need it for practical use in perl so.. ^^)

To simulate this problem I've written the following code: 
print 'num of splits [positive integer]: ';
my $splt = <>;
print 'chance of going deeper [between 0 and 1]: ';
my $chance = <>;

my $avgOf = 1000000;
my $sum = 0;
for(my $n=0;$n<$avgOf:$n++)
{$sum += dep();}
print "avg depth is ".($sum / $avgOf)."\n";
exit;

sub dep
{
  if(rand() < $chance)
  {return 1+max(multiDep())}else
  {return 0;}
}

sub multiDep
{
  my @keep;
  for($i=0;$i<$splt;$i++)
  {$keep[$i] = dep();}
  return @keep;
}

sub max
{
  my $hi = shift;
  for my $v (@_)
  {if($v > $hi){$hi = $v;}}
  return $hi;
}

[download]
I need to be able to tell the average value returned by "dep()" (printed number), as a function of the given value of $chance.
How can I calculate (mathematical function) it when:
'$splt = 1" "$splt = 2" "$splt = 3" "$splt = 4" etc

I figured the question for the "$splt = 1" case, where the desired formula is $chance/(1-$chance).
The solution required knowledge of:
basic statistics, basic geometric series, basic integrals.

I would go by experimenting than graphing and than theorizing, but even when only examining the average of 1000000 tries each time, the result still varies too much to be considered accurate enough for this strategy.

I'm sure if I can understand the solution for "$splt = 2" I'll be able to figure the rest by myself.

Help please ^^
Thx a lot

In reply to Average random depth formula required by palkia

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