Is this a correct implementation/interpretation of the "Birthday Problem"?
Nitpicks

$chars = '1234567890qwertyuiopasdfghjklzxcvbnm';


I would find this easier to read if it were in alphabetical order.  

$chars = '1234567890abcdefghijklmnopqrstuvwxyz';


That way I can see that are no letters missing.  

$alphanumeric_alphabet = '0123456789abcdefghijklmnopqrstuvwxyz';


I think that makes it clearer what the string is.  Also, I prefer to name scalars (numbers, strings, etc.) with singular names.  I'd name $combinations $combination_count for the latter reason.  

$probability = 1.0;


This is the probability that you can select the first combination without a collision.  

foreach(range(1,$count) as $i) {


This is neat, but PHP doesn't handle it well.  It will create the whole array and then iterate over the array.  

for ( $i = 1; $i &lt;= $count; $i++ ) {


This is more efficient in PHP.

Mathematics

However, 

for ( $i = 1; $i &lt; $count; $i++ ) {


is probably more correct.  Remember that the probability of finding a combination without a collision is 1.0.  So the first iteration through the loop is the 2nd, and the $count iteration is the $count + 1 combination.  

Now, if instead of the birthday problem, you are looking for the chance of a match to an already existing combination, then this would be closer.  However, that problem would normally not care about collisions in the match tries.  So it seems you should either change the upper limit in the for loop or swap out the loop for 

$probability = pow(1 - 1/$combinations, $count);


That's the chance of not matching a preexisting selection with $count tries.  Note that in that case, the tries can match each other.  If you want the probability of a collision:

$probability = 1 - pow(1 - 1/$combinations, $count);


More Nitpicks

Assuming the birthday problem is the right model,

    $probability = $probability * (1 - ($i/$combinations));


You could write this as 

    $probability *= 1 - ($i/$combinations);


Which seems simpler.  Note that the parentheses around the division are unnecessary as division is higher precedence than subtraction.  Not normally a big deal either way.  They do slow down the loop slightly though, so if you have performance issues, you could try removing them.  With high enough iterations, it could shave off noticeable time.  

echo "A total probability their being collisions of ".($probability*100). "%\n";


You misspelled "there":

echo "Total probability there being collisions of ".($probability*100). "%\n";