binomial distribution unfair coin

Question

I'm working on two bionmial distribution questions about unfair coin, where I got confused for one of em by its wording.

Unfair coin: You have a coin with which you are 2 times more likely to get heads than tails. You flip the coin 100 times. What is the probability of getting 20 tails? What is the probability of getting at least one heads?

for 20 tails:

P(x=20) = 100!/(20!)(80!) * (1/3)^20 * (2/3)^80

anything wrong here?

for 'at least one head':

This is where I feel little lost. do I get the sum of probability from getting 1 head to getting 100 heads? Is there any other trick to find the answer?

Answer 1

Your expression for $P(x = 20)$ looks good to me.

Hint For the other question: The complement of the event "at least one head" is the event "no heads".

Answer 2

What you can do is calculate the probability there is absolutely no head, then do one minus the probability of no head. $$P(\text{no head}) = \left(\frac13\right)^{100},$$ so $$P(\text{at least 1 head}) = 1 - P(\text{no head}).$$