Steps to reproduce:

1. Install 64-bit 6u10 JRE
2. Launch 64-bit 6u10 JDK installer, it will still offer the public JRE (it shouldn't)
3. Keep the public JRE feature selected, and proceed with install.
4. JDK installer will say that you already have a JRE installed, would you like to reinstall.
5. Select YES.
6. JDK installer will uninstall JRE, but not reinstall it.

Expected behavior: Actually, as of 6u10, the public JRE should not be offered if they already have it. So step 2 is the bug.
See also CR 6712020: jdk 64 bit Installer doesn't offer JRE if 32 bit JRE is already installed in the system.
Same cause, different test case.
See also 6697772. Re-assigning this CR to Chris as it should be addressed with his fix.