I'm rather new to php and i am stuck.
i want to have a single checkbox be calculated with php, which i partially get done, but when the checkbox is not selected, php does not recognize the cb1 index with $_POST["cb1"]. When the checkbox is selected, no errors are given.

Well I think thats a correct behavior..when your checkbox is checked it'd surely work fine. When checked, the variable name will contain the value of the checkbox. Look at the somewhat simimlar samples at the below links:-
http://www.globalissa.com/articles/articleCheckbox.php
http://www.smartwebby.com/PHP/Phptips2.asp

If your check box markup looks like:
<input type="checkbox" name="myCheckBox" value="Yes" />

The check box "mycheckbox" would have a value in the post array only if the checkbox is checked. Otherwise the value would be nothing.  so ensure that you check if the value is set and to checked. You can do it like:

<?php
	if(isset($_POST['myCheckBox']) && $_POST['myCheckBox'] == 'Yes') 
	{
	    echo "checked.";
	}
	else
	{
	    echo "not checked";
	}	
?>