Here’s what I’ve been taught:
Scenario No. 1: BG holding hostage with knife to neck or side of head. You enter, draw and take aim at BG’s head. No need to say anything, but if you say something (perhaps in response to direction to “drop your gun”) you say:
“This ends one of two ways. Draw one drop of her blood, and you’ll be dead before the blood hits the floor. Let her loose and you live.”
Of course, there is a third possibility, namely that while you and BG are standing off, you get and take a clear shot. However, this possible outcome should not be presaged.
Apparently there are statistics that lead to the conclusion that in the vast majority of cases, the BG releases the hostage, once he registers that you are not going to drop your weapon or lower your aim.
Scenario No. 2: Same as above, but BG armed with pistol, pointed at hostage’s head. The difference between the scenarios is that in No. 2, BG is as much a threat to you as he is to the hostage.
Enter, draw and take aim as above. Speech (if appropriate) the same. However, key to this scenario is to stay focused on BG’s gun. Here statistics say it is 50-50 (once he registers that you are not going to drop or lower your weapon) that BG will move his aim from hostage to you, in which case you have no choice but to take the shot.
Note that scenario does not end if his grip on hostage is released; this action almost always precedes BG’s taking a shot at you. As a result, you are statistically better off to take your shot the instant that hostage is out of the line of fire.
In either case, never surrender your weapon. Statistics say that this virtually assures that one or both of you and hostage will die.
I’ve never been in this situation, and I’ve done no independent research of statistics. Likewise, the layout and security systems of my house make the scenario there unlikely (though not impossible). However, I trust the folks from whom I acquired the foregoing, and it constitutes my “plan.”
As prior poster sagely observed, in all of the foregoing remember to stay observant and flexible. No plan applies to all cases.