I'm assuming that #select_1 is to have the chosen plugin applied to it. If this is correct, #select_1 is going to have a style of display:none; applied to it, so you not going to register any 'click' events on it.
If you look at the "chosen" documentation, http://harvesthq.github.io/chosen/, you can discover how to register a change handler to your chosen element.
If chosen is applied to #select_2 and #select_3, then toggling the visibility of these two elements is pointless because they are already hidden. The HTML rendered by the chosen plugin will have container elements with IDs of #select_2_chosen and #select_3_chosen - so toggle the visibility of these!
The code below assumes that #select_2 and #select_3 have the class 'chosen'. I am applying the chosen to #select_1 separately so I can bind the change event that is to be unique to it. Finally, I trigger the change event immediately after defining the change handler so that #select_3_chosen will become hidden.
$(".chosen").chosen();
$("#select_1").chosen().change(function () {
var value = $(this).val();
if (value=="A") {
$("#select_2_chosen").css('visibility','visible');
$("#select_3_chosen").css('visibility','hidden');
} else if (value == "B") {
$("#select_2_chosen").css('visibility','hidden');
$("#select_3_chosen").css('visibility','visible');
}
}).trigger('change');