Anti-aircraft fire wasn't terribly accurate. In The Bombing War: Europe 1939-1945 Richard Overy gives some figures for British defences during The Battle of Britain and The Blitz: over the whole night campaign from July 1940 - June 1941 a massive number of shells were fired (260,000 in September alone) in return for which 170 aircraft were claimed shot down and perhaps 118 damaged. Air Ministry estimates were that it took 6,000 shells for each claim in autumn 1940, improving to 3,195 in April 1941. New searchlights with radar detection, more guns and better organisation improved matters by spring 1942, reducing the shells per claim to 1,830, the lowest of the war.

Overy also quotes a figure of 29,000 rounds fired by AA guns around Moscow on the night of 21-22 July 1941 against German aircraft, with German records showing a single aircraft lost that night. Edward Westermann's thesis on German Ground-Based Air Defenses, 1914-1945 gives a German estimate of 3,000 rounds per shootdown in 1943.

Ack-ack guns were particularly inaccurate when used for barrage fire rather than aimed fire, but before radar guidance was widely available they were sometimes used in this capacity for morale more than military purposes. Westermann does point out that the effectiveness of flak was not solely measured in aircraft brought down, though; it forced attackers to fly higher and perform evasive manoeuvres, reducing the accuracy of bombing, and was responsible for considerable numbers of aircraft damaged.

As for how friendly aircraft avoided being hit, generally AA guns were employed in zones that friendly aircraft would try and avoid; [this map] (https://www.ibiblio.org/hyperwar/UN/UK/UK-RAF-III/img/RAF-3-07-01.jpg) for example from The Royal Air Force, 1939-1945, Volume 3: Fight is Won shows British defences against V-1 flying bombs in 1944 that started with three patrol lines of fighters before a belt of guns in front of London, then evolved into a coastal gun belt with fighter patrols before and after. Friendly fire was always an occupational hazard, though, especially as gunners tended to shoot first and worry about aircraft recognition later (the famous black and white stripes worn by Allied aircraft around Overlord were an attempt to assist with this; almost 800 volunteers from the Royal Observer Corps also served on ships to help with recognition). Westermann has a table of Luftwaffe fighters shot down by friendly fire over the Reich in 1943, between 10 and 35 each month for a total of 229.

Frank Zeigler, intelligence officer of 609 Squadron, wrote about his time with the squadron in the RAF Flying Review. When first introduced the radiator scoop of the Hawker Typhoon often caused it to be mistaken for the Focke Wulf 190, drawing fire from coastal gunners. Zeigler writes: "Once a gunnery officer, whom I had just "blitzed" on the phone, actually called back with the request: "Could you ask your pilots tactfully - very tactfully - did we get anywhere near them?" Sometimes they did. Roy Payne, wading ashore after being hit and crash-landing in shallow water, really lost his temper when the coastguards addressed him in German."

(Addendum: another anecdote from Under the Bombs: The German Home Front, 1942-1945 by Earl R. Beck on the lack of accuracy of AA guns. A joke from the time concerned a soldier who was condemned to death but allowed to choose the method of execution. He elected for death by anti-aircraft fire, so was put in a tower surrounded by AA batteries that opened fire. Three weeks later he finally died... from starvation.)

I can't give your the statistical number crunch but I can provide some insight into the working of the German AA/American aircraft.

First off, there's two different kinds of AA fire:

• High caliber timed fuze shells that explode nearby enemy planes. It's got a slower rate of fire but is effective at higher altitude.

• Smaller caliber, rapidly fired shells that explode on hit.

The former would be used a lot against your average bomber raid, which was flown at ~27 000 feet. Obviously the shot has to be led in order to hit, but wasn't just done by feeling but by calculation.

The optics were to track the gun, determining the altitude, angular height and direction. Radar could also be used to track the plane. This information would be fed to a mechanical computer kit that computes the necessary fire direction. A large concentration of AA batteries (all with their own mechanical computer kit) would then light up the concentration of bombers.

Of course the Allies weren't content to just let the German AA take them out; the easiest way to mess up any German attempt was to simply start flying in a different direction when they were picked up by German AA. The gunners would have to recalculate every time the plane would fly in a different plane. The American doctrine was to fly an irregular pattern with changes in direction (minimal 20 degrees) and height (minimal 500 meters) every ~30 seconds when they approached a defended area.

Follow up question.
I hear that it was not uncommon for AA fire to not detonate in the air and fall back to the city doing a fair bit of damage. Is there any truth to this ?

Follow up question: has there been anything written about damage from shrapnel falling back to the ground?