Geometric distribution example

In the second case why don’t we give the ans as 1-P(X≥11) which goes as 1-p(p-1)^10 rather we give the answer as 1-(1-p)^10. or since these are disjoint events why dont we just sum up the probabilities of all pX(x) for x in {0:10}?

For the case used in the answer, please refer below explaination:
Success here is determined by probability of finding a donor, in atleast one of the first 10 volunteers.
In the case

It is one of the correct solution, but the equation written by you is for 1-P(X=11).
We’ll have to calculate it upto population, hence becomes much expensive computationally.

Whereas, in the following case:

You missed out the inclusive principle, i.e. what if more than one volunteer has a matching blood group? As the question asks ATLEAST.
Hope this helps.

Thank for the solution, understood the first part of it. In the second part when i said x in {0:10} i meant x from 0 to 10, as in summing up the probabilities of r.v from x=0 to x=10.

Yes, but we’ll also have to include inclusive probabilities, example: if x =1 and x=2 both are in the case.