Firefox may be looking in the wrong location for the Silverlight plugin if it doesn't find that new version.

You can see the installed plugins on the about:plugins page.
You can set the pref plugin.expose_full_path to true on the about:config page to see the full path of plugins on the about:plugins page.
It is best not to leave that pref set to true as it exposes that full path to web servers, so reset that pref to false after you are done with the about:plugins page.

You can open about:plugins and about:config via the location bar, like you open a website (about: is a special protocol to access some build-in pages).
If you get a warning when opening the about:config page then you can confirm that you want to continue.

• Troubleshoot issues with plugins like Flash or Java to fix common Firefox problems

Even when I set the expose full path to true, Firefox still shows that it is running Silverlight 3. When I use the Mozilla plugin check, it also indicates that I need to update from Silverlight 3 to 4. Your answer really did not solve my problem. I have tried multiple times to update to Silverlight 4, but to no avail. If Firefox is indeed looking in the wrong place for the Silverlight update, how do I get it to look in the right place? As I said before, Safari doesn't seem to be having any trouble, but I would rather use Firefox as my browser.

Try to remove the Silverlight 3 plugin files from the location that shows on the about:plugins page.

If Firefox doesn't find the Silverlight 4 plugin then copy the plugin files to the same folder where you originally had the Silverlight 3 plugin.