The trigger for the set session value is wrong.
It is set to " if session does not exist " which means it is created as zero the first time someone visits the page and is not updated.
Instead use the lightning bolt and trigger off of the radio button itself. It will look like:
<?php
if (!session_id()) session_start();
if(((isset($_POST["choix"]))?$_POST["choix"]:"") != "") {
$_SESSION["choix"] = "".((isset($_POST["choix"]))?$_POST["choix"]:"") ."";
}
?>