try this
if (choice == 1){
//$('#checkbox').change(function(){ you are callinjg a change event here... so the hide is not fired..
$('#hide').removeClass('hidden');
// });
}
UPDATED
I want the div with the id=hide to appear when someone selects the second option in my dropdown menu (selectedIndex of 1), and disappear again if they select a different option.
i used the change event of the selectbox.. and checked the selected value.. if value is your second options's value.. then remove class else addclass
$("#product-select").change(function(){
var choice = $(this).val()
if (choice == '269270608'){
//$('#checkbox').change(function(){
$('#hide').removeClass('hidden'); //remove class here
//});
}else{
$('#hide').removeClass().addClass('hidden'); //else addclass
}
});