What does independent mean here?

That one event happening tells you nothing about the other. Rain and a train delay are usually not independent — rain causes delays — and for events like those these formulas understate the joint probability.

Why is "neither" not just 100% minus "both"?

Because between them sits the case where exactly one happens. The four outcomes — both, only A, only B, neither — are what add to 100%, not just the two extremes.

Can I chain more than two events?

For all of them happening, yes: keep multiplying. For at least one, it is easier to work out the probability that none happen and subtract from one, which avoids counting overlaps at all.

Math

Probability Calculator

Enter the probability of each event to see how they combine.

The link carries your figures, so it reopens on exactly these numbers.

The formula

both = A × B

either = A + B − (A × B)

neither = (1 − A) × (1 − B)

exactly one = A + B − 2 × (A × B)

Worked example

A 50% chance of rain and an independent 30% chance of a delayed train

0.50 × 0.30 = 15% both
0.50 + 0.30 − 0.15 = 65% at least one
0.50 × 0.70 = 35% neither

Where this goes wrong

Adding probabilities that can both happen

A 50% chance and a 30% chance do not give an 80% chance of at least one. Adding counts the overlap twice, so you have to subtract the 15% where both occur, leaving 65%. The giveaway is when the sum passes 100% — two 60% chances cannot make a 120% chance of anything.

Questions

What does independent mean here?: That one event happening tells you nothing about the other. Rain and a train delay are usually not independent — rain causes delays — and for events like those these formulas understate the joint probability.
Why is "neither" not just 100% minus "both"?: Because between them sits the case where exactly one happens. The four outcomes — both, only A, only B, neither — are what add to 100%, not just the two extremes.
Can I chain more than two events?: For all of them happening, yes: keep multiplying. For at least one, it is easier to work out the probability that none happen and subtract from one, which avoids counting overlaps at all.