If I’m in the movie, i do what the script/director demands.
If it’s real life, I do what the hostage taker demands.
Focus on the bridge of the Bad Guy’s nose. Aim carefully while saying “I don’t want anybody hurt here. Why don’t we <blam>…”
Apparently, there’s a part of the brain that, when hit with a bullet, results in immediate incapacitation. Flaccid paralysis (shooting) - Wikipedia
I’d only try to take that shot if 1) I had extensive training for that type of task and the situation looked to be worsening 2) the hostage taker looked like he was imminently about to kill someone.
Other than that, a minimally rational hostage taker has to realize that if he kills the hostage, his leverage disappears and I’m free to shoot. I suppose a hostage taker might merely injure the hostage to incentivize me to comply.
In any case, I’m not dropping my gun. Whatever his goal may be, the situation won’t get better if he’s the only one armed.
Since the handgun of my choice and spec has an integrated laser, then yes. 
If you’re going to retcon that option out, then I’m most likely to go with the Keanu Reeves option of shooting the hostage in the leg and then the perp center mass.
Is it true there’s a place in a man’s head, where if you shoot it, it will “blow up”?
Have you ever jumped through a doorway firing two guns while screaming argh?
I don’t think there’s much pent-up tension in the head for genuine blowing up to happen. Perhaps “shatter and splosh” is more accurate, like this: https://www.youtube.com/watch?v=xEIvPD6zpL0 and it would have to do with caliber, projectile type, range and angle of entry.
If you apply enough kinetic energy to someone’s head in a short enough amount of time, it will behave like a watermelon hit with a baseball bat but that has little to do with the precise spot. You can find evidence of this on 4chan and liveleak if you’re doubtful although my sympathetic recommendation is to take my word for it.
No.
It was a Hot Fuzz reference.
It did sound like a joke but I figured I’d answer it straight anyway.
I presume it was from Constable Butterman (Nick Frost)?
yep!
Yes, if you shoot it with a 25mm canon.
Nonsense. Their copiers can’t fire a bullet worth shit. 
ETA: Neither Pachalbel’s nor Tallis’ canon were a 25mm canon. I checked. 
How well do I like the hostage?
I take the shot. Either I kill the bad guy and am a hero or else I kill the hostage and THEN kill the bad guy and blame it all on him.
“Don’t ask me, his head was like that when I got here…”
I don’t know if there’s some sort of policies and procedures manual here that we could refer to. However, game theory should be pretty clear that a best option here is to play for time. Stay in the standoff and wait for a variable to change.
- The bad guy shoots the hostage. Decidedly sub-optimal but at that point he’s awaiting death or arrest.
- The bad guy gets talked off the ledge. He gives up and things de-escalate.
- The hostage changes the situation somehow. Fighting, fainting, what have you. This opens an opportunity for good guy.
- Other law enforcement arrives, thereby changing the calculus.
- The standoff continues
It just appears that taking a close-held hostage is not an ideal situation from the bad guy’s perspective. Were he good at what he does, he’d have gotten away with whatever with no one knowing at the time. Taking a hostage seems to be a things-have-gone-wrong low-probability outcome.
Given the fact I have shot a gun in over 50 years, I wouldn’t shoot.
If I had, and was a crack shot … I wouldn’t shoot, either. The chance of the hostage getting out alive is far worse (and even a crack shot can miss the target and hit the hostage).
One question I have for everyone who is saying they would take the shot.
If the hostage was your child, would you still shoot?
Yes, cops never drop their guns or hand them over in this sitrep. They continue to cover- they rarely do take the shot, but they wait for a opportunity. Time is on their side.
I consider myself a pretty good shot with a handgun, but I’d take Jonathan Chance’s approach and play for time. If he’s not moving (getting away) and he’s not actively harming people, additional time is probably a benefit to me and the hostage.