There are 2 mistakes in your code which will not make it work:
document.ready(function () {
it should be
$(document).ready(function () {
The other thing is that the button is a submit server side button, so even after the p tags are hidden, after the load of the page, everything will be loaded again. Hence for this to work, you will have to prevent the server side eventing.
Check the corrected code below:
<script type="text/javascript">
$(document).ready(function () {
$('#' + '<%=btnSubmit.ClientID %>').click(function () {
$("p").hide();
return false;//This important to avoid reload of the page again, which will again make the <p> tags visible.
});
});
</script></p>