Besides, Firefox doesn't really use the JRE. It is merely using a Java plugin to call up the JRE in order to run Java applets on websites. So you really need two things to enable Java support for Firefox, or any other browser. You need to have either the 64-bit JRE or the 32-bit JRE. Then you need the appropriate Java plugin for the browser.
Now if you went for the 32-bit JRE, and the Java plugin, you would have Java applet support in your browser, but you would only have limited Java application support. The limitation being that you would not be able to run 64-bit Java applications on your computer.
But if you go for the 64-bit JRE and a Java plugin for the browser, you could run both 64-bit and 32-bit Java applications, as well as Java applets. Correct me if I'm wrong but 64-bit JRE can run both 64-bit and 32-bit Java applications.
Now, when you go to
www.java.com there is no special link that would allow you to choose what you want to download, if you want the 32-bit or 64-bit JRE. There is only this stupid "download" button which doesn't tell you a s***. You know? So the only way to tell really what you are getting is to head for the JDK, which is obviously available in both 32-bit and 64-bit, and they include a JRE of the same edition.
Now I did install the JDK 64-bit, therefore I know that I have the 64-bit JRE. But installing the JDK didn't seem to install the Java plugin for Firefox, only for Internet Explorer. So I had no choice but to go visit the
www.java.com from within Firefox and hit the "download" button. You need to visit the page with the browser for which you intend to install Java, so that you get the appropriate plugin version.