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I feel like I have read this information before somewhere, but I cannot find it anywhere.
In my campaign, I am planning an "ambush." I use the quotes because the party's scout will be more than up for the challenge of discovering the ambushers.
So, the reality is that my players' party will be fully aware of what they are heading into. Thus, they will be able to cast any and all buffs before the fight begins.
So, what is the appropriate CR for an encounter that will be the only encounter for an entire day for a party of 4 level 4 PCs?
They can probably handle a CR 8 or 9, possibly even 10 encounter, but it depends on a lot of things.
If there is a chance that they will stumble blindly into it then do not go above APL+4. If they will be able to buff minute/level spells prior to combat then they can probably handle CR 11.
Don't use individual creatures above CR 6 because the party will have trouble dealing with its defenses (SR and DR) and its DCs are likely to be too high for the party to cope with.
Indeterminable.
Depends on what's doing the attacking, and the capabilities of the PCs. Add into that the fact that the CR system is, at best, incredibly inaccurate and it is simply impossible to state.
Technically speaking, a party of four Xth level characters is a CR X+4 encounter and, because it's a mirror image of the party, there is a 50/50 chance of winning or losing.
The rest is intelligence, preparation, tactics, and luck. For example:
* If one side has a chance to buff and the other side doesn't, the buffed side will probably win.
* If one side thinks they may have to save resources for later in the day, the side that knows this is a go-for-broke 5 round adventuring day will probably win.
* If one side uses synergistic tactics that enable the group to work together effectively, the synergistic side will probably win.
* If one side knows about the other side's probable tactics and weaknesses (and can capitalize on them), the more knowledgeable side will probably win.
* If the terrain/environment favors one side over the other, the favored side will probably win.
et cetera, et cetera, and so on.
Depending upon circumstances, each of those factors could be worth anywhere from nothing on up to 2-3 CR; knowing nothing about the group, I'd be inclined to rate each of them at roughly CR 1 each. So if everything breaks the PC's way, a 4th level group could have a 50/50 shot against CR 12-13.
However, I've never seen everything go the PC's way.
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Well, it's okay for the encounter to be tough. After all, the party has a whole day's resource budget to burn. Although they might not know that; maybe they hold something in reserve just in case there's another encounter that day.
However, don't be deceived by APL vs. CR; it's not that simple. A 6-PC party is +1 APL because they have a lot of action economy. But against an enemy of CR = APL+4 they'll maybe do worse than a 4-PC party would, because the CR vs. individual PC power difference is wider. A single enemy that's 4 points of CR above the party is in truth 5 CR above the level of the PCs in a 6-PC party. That might mean that none of the PCs can really hurt the monster, and that the monster will just kill a PC every round.
Nominally, two CR 7 monsters present the same encounter level as a level 9. But for the APL 5 party, the first encounter is probably more survivable than the second one. They might not be able to hit the CR 9 monster, but they probably have some way of focus-firing one of the CR 7 monsters, dropping it fast and then concentrating on the second one. Their defences won't hold against the CR 9 one, but they can hold out against the two CR 7s for a couple of rounds, enough to swing the battle in their favor.