"A bag contains equal numbers of red , green ,and yellow marbels. If Geeta pulls three marbles out of the bag, replacing each marble after she picks it, what is the probability that at least one will be red?

I understand the guide's solution:

Assuming there is one of each marble in the bag, there's a 1/3 chance of getting a red and a 2/3 chance of NOT getting a red on any given try.

P(r>=1)=1-P(r<1)=1-P(r=0)=1-(2/3)(2/3)(2/3)=1-(8/27)=19/27

What I don't understand is the more cumbersome method and why doesn't it equal 19/27?

My thinking: P(r>=1)=P(r=1)+P(r=2)+P(r=3)=(1/3)(2/3)(2/3)+(1/3)(1/3)(2/3)+(1/3)(1/3)(1/3)

= 4/27+2/27+1/27

= 7/27 (DIFFERENT than 19/27)

Can someone please tell me what I'm messing up here? Thank you in advance.