[CentOS] Confusion about 32 vs 64 bit jre

I ran a yum update yesterday and got the new kernel and other goodies,
and I've rebooted and all seems to be well.

Except, I just tried to run Firefox for the first time, and it wants a
new jre (6.16).  Now that's fine, except that I have a recollection of
needing the 32-bit jre the last time I did this, but that's not
working.  I'm pulling down the 64-bit version, but I'm wondering if I
need both, which, IIRC, does not work, or which one of the two?

(I _am_ running the x86_64 OS and most apps.)

What bothers me is that I've tried both, and apparently neither one
works (!!!!).

I installed by downloading the bin file (self-extracting rpm), and
then running the bin with sudo.

Where did I go wrong?

mhr