consolidating terms with conditional probability

Question

problem:

A coin is biased so that P(H) = .6. Toss it 10 times. Find the probability of at least 9 heads given that you got at least 8 heads.

answer key (in part):

P(at least 9H | at least 8H) = $\frac{P(at\ least\ 9H\ and\ at\ least\ 8H)}{P(at\ least\ 8H)}$ = $\frac{P(at\ least\ 9H)}{P(at\ least\ 8H)}$

I don't understand how the numerator simplifies here. If anything, I would have expected P(at least 8H) since P(8H or 9H or 10H) includes P(9H or 10H).

Answer 1

P(8H or 9H or 10H) includes P(9H or 10H)

or more precisely

(8H or 9H or 10H) includes (9H or 10H)

Yes, and therefore saying that both occur simultaneously is the same as saying the second occurs.