At sea, there are two basic types of attack: torpedo and missile. For all intents and purposes, gun battles between surface ships will not occur barring extremely unusual circumstances.
Missile attack relies on launching missiles from surface vessels, aircraft, or submarines. These missiles come in two basic flavors: supersonic missiles that come straight at you at high altitude and low-flying cruise missiles that hug the water to avoid detection until the last minute.
In either case, the defender needs to stop the missiles before they strike his ships. Modern warships essentiall eschew armor in favor of active defenses; while many have some limited degree of protection against near misses, terrorists with machineguns and the like, and really large ships like carriers have thick hulls just by virtue of their size, modern missiles would pierce old-style steel armor plate with relative ease, mainly due to the advent of shaped-charge warheads. This is not to say armor is useless; just that the amount needed to actually keep out modern missiles or torpedoes would be unreasonably heavy and expensive, and even if you could much of what ship needs to fight (radar antennae for example) cannot be armored.
Therefore ships rely on measures such as ECM and decoys to lure missiles away, and various forms of air defenses to destroy them. The ideal with any missile is really to shoot down or sink whatever is launching it before it can fire. Failing that, a ship must destroy or avoid incoming missiles.
Any formation of ships (even if its only one ship) will be able to stop a certain number of missiles. In actual practice the vagaries of life will adjust this number up or down but a pretty good idea can be had based on what's generally available to the formation to defend itself based on the average success rate of different systems. This is further adjusted by how long the ship has to engage the incoming missiles; really fast high flying ASMs can be seen and fired at a long way off, but they come in really fast. Sea-skimmers, on the other hand, come in slow, but cannot be detected until they are much closer.
What this means is that each defending ship in a formation will be able to launch a given number of salvos of SAMs in the time avilable and each salvo will contain a given number of SAMs based on the number and type of launcher. A certain percentage of these will hit. Furthermore, a quick-and-dirty assumption can be made that each ship will, if it contains the appropriate system, defeat 1 missile with guns (dual-purpose and point defense) 1 with ECM and 1 with decoys, assuming they are trgeted on the defender and not some other ship; i.e. an escorting destroyer can absorb 3 missiles targeted on itself with guns, ECM and decoys, but if there are none then 3 missiles targeted on merchant ships cannot be engaged with those systems instead.
Take the following diagram (I didn't draw this myself; I found it elsewhere on the web)
This example shows a carrier battlegroup including 4 Arliegh Burke Aegis destroyers, 2 Oliver Hazard Perry frigates hypothetically upgraded with Evolved Sea Sparrow to replace the now-removed SM-1 missile system, and a carrier and ammunition ship each with Evolved Sea Sparrow and RAM missiles. Each ship can be assumed to have PD guns, ECM, and decoys. Note that the smallest range circle, the red for RAM is still a 16 KM radius as opposed to about 2KM tops for PD guns. While this is suspicious (other sources show only a 7.5km range) it still confers a major advantage over guns. Note also the E-2C Hawkeye AWACS track above the formation, and the direction of air threat.
As an aisde, the destroyer that is out away from the formation is acting as a "missile trap"; it would remain with radar shut down untilt he Hawkeye detected the bombers within range of the destroyer then it would suddenly activate and fire SAMs at the bombers which will, for obvious reasons, try not to get too close to the ships they are attacking. The frigate out ahead of the formation is sweeping for submarines. Any friendly submarines attached to the formation are left off as they are not relevant to the air defense picture.
Let's assume for example that this formation is attacked by bombers armed with AS-6 Kingfish missiles. This 13,000 pound monster has a speed of about 3400km/h (mach 3.5 depending on altitude) with a 1000kg warhead (or a 350kt nuclear warhead); hence the importance of avoiding even one hit and the pointlessness of trying to stop such a thing with armor.
A bomber really could not be expected to carry more than two, and possibly only one depending on range to the base. In either case, the bombers will doubtless be detected at well beyond maximum range by the E-2C hawkeye, or by any radar or other sensors they use to locate the ships. The missiles will run in at about 65,000 feet until they are almost over the formation, leaving them in full view. The bombers will launch at maximum range in order to limit the amount of time they are exposed to attack by the carrier's fighters.
Maximum range for the missile is given as anywhere from 300 to 700 km, but the shorter range is proably safer due to the need to locate the ships before firing. Therefore, at 3400km/h it will take the missiles 4.8 minutes or 288 seconds from launch to arrive at their targets. The ships have this long to destroy them.
An Arleigh Burke carries a 30-cell and a 60-cell VLS. For purposes of the example, we will assume that each Burke has 20 of these cells loaded with Tomahawks for antiship or land attack and therefore not available for air defense, and 10 more loaded with ESSM for closer-in. Each Burke will be loaded with SM-2 Block IIIB SAMs; with 4 Burkes this gives a total available SM-2 load of 240 missiles. Since each Burke has a 60-cell and a 30-cell launcher, (we will assume the tomahawks are evenly distributed between the two) and each launcher can fire one missile per second, that allows 8 SM-2 launches per second for 20 seconds and 4 per second for an additional 20 seconds from the formation as a whole (despite the illustration, we'll assume the missile trap destroyer is actually back with the formation to keep this simpler).
This version of the SM-2 can engage at 170 km, and launchers will actually be managed so that the first wave of SAMs arrives just as the AS-6s break this barrier. At this point, the AS-6s are 163 seconds from impact. 8 more SAMs will arrive every second for 19 more seconds, then 4 more per second for 20 seconds. We will give each SM-2 an 80% chance of a hit (not at all unlikely since this is what AEGIS was intended for back in the 1970s and both it and the missiles have been upgraded considerably). The SM-2s can be expected to stop 192 incoming missiles by the time 40 seconds have passed. This is the beauty of the AEGIS system; it does not have mechanical illuminators that need to be steered onto targets. Although not all the missiles can be directly controlled at once, they don't all have to be since they are arriving in waves, and there are 4 AEGIS systems to control the missiles.
If there are any missiles remaining, the carrier, the 2 frigates and the ammunition ship can engage with Evolved Sea Sparrow as can the destroyers. The ESSM can be quad-packed in the VLS systems of the
Burkes, meaning that the 10 cells we devoted to these can actually carry 40 missiles. I cannot locate a rate of fire for the non-VLS launchers of the other systems, but we will assume the average launch rate of 1 per 2 seconds that older mechanical launchers used, and still 1 per second for the Burkes. ESSM has an effective engagement range of 50km, and whiel the non-Aegis ships actually do have illuminators and directors, they can also just let the AEGIS ships control their missiles. Allowing for 1 launcher per frigate, 2 for the carrier and ammunition ship each, this allows an average rate of fire of 11 ESSM per second starting at 50 km. At 50 km the AS-6s are 48 seconds from their targets. It is safe to assume, however, that the ships will have no more than 20 seconds worth of ESSM available (the capacity of the destroyers); this still allows a total of 220 ESSM to be launched.
Because not all the ships are on top of each other and the missiles will go for different targets, it will be very hard to get hits with the last ESSM launches. We'll take an overall average of 50% hits for ESSM. This allows 110 hits.
At this point, RAM, gun systems, ECM, and decoys take over. RAM, assuming 1 launcher per escort and 2 for the 2 large ships means a total of 10 launchers firing 1 missile every 2 seconds, and we can expect a 50% chance of a hit. With the AS-6 covering 0.94 km/s from 10 km out can expect 50 RAM launches and 25 kills.
This gives us a total of 327 AS-6 kills. Each ship can be expected to defeat another 3 missiles targeted specfically at it, but these are not totalled in because the missiles will generally tend to go for the bigger targets.
This means that, to get at least one hit, a bare minimum of 331 missiles would be needed, or 166 bombers' worth, and that assumes that both A) the bombers are operating close enough to base to carry 2 each and B) totally ignores missiles or bombers destroyed by F/A-18s from the carrier. If land-based fighters were nearby or worse, a second carrier, the number gets truely ridiculous.
Keep in mind however, that this is based on an AEGIS-centric defense, and China does not have AEGIS. Moving on to torpedo and submarine attacks next, before we get into the actual situation in the pacific.