Is the chance of independent events just summed, when calculating overall chances of any event?

Question

I'm not a mathematician or statistician (my last exposure was over 30 years ago with stats classes in school).

My problem is if events are independent do you just add them together to get the overall chance of the event (my school exposure at least makes me think that maybe it is not that simple).

The actual problem:
I'm about to have a heart operation.
There are chances of serious complications.
1% risk of A (cardiac perforation)
1% risk of B (stroke)
1% risk of C (pulmonary vein stenosis)
1% risk of D (vascular damage)
0.2% risk of E (mitral valve damage)

So, is my chance of a serious complication 4.2% (A + B + C + D + E)?
Or since each event has no influence on occurrence of other events is chance of a complication more like 1% being the highest risk value?
Or something in between?

I did try searching here for an answer but everything rapidly got into stats maths and complex formulae, then I got lost, so please keep it simple for me   <:-)

Answer 1

If you're asking "What is the probability of at least one serious complication?", then you don't add the probabilities.

(Consider: if there were 35 complications that each occurred 3% of the time that would imply that your chance of no complications was less than zero!)

If you assume the occurrence of the various kinds of complications are all mutually independent (which would seem unlikely to me), then:

You work out the probability that there are no complications, and subtract that from 1.

That is

$ P(\text{at least one complication}) = 1 - P(\text{no complications})\\ = 1 - P(\text{no complication A})P(\text{no complication B})\ldots P(\text{no complication E})\\ = 1 - (1-P(\text{A occurs}))(1-P(\text{B occurs}))\ldots(1-P(\text{E occurs}))\\ $

In your example:

1% risk of A (cardiac perforation)
1% risk of B (stroke)
1% risk of C (pulmonary vein stenosis)
1% risk of D (vascular damage)
0.2% risk of E (mitral valve damage)

If we assume these to be independent (as I said, somewhat dubiously), the chance of at least one complication is $1 - 0.99 \times 0.99 \times 0.99 \times 0.99 \times 0.998 = 4.13\%$

If the probabilities are all small and there aren't too many of them, it's almost the same as adding, but a bit less.

However, there will be a tendency for problems to occur together (some of those other complications would be associated with higher risk of stroke, I'd imagine) - and that lowers the chance that there will be any complications (while increasing the chance of more than one of them). So in general, the chance of some complication will tend to be lower than what you work out via the independence assumption.