Importance sampling a rare coin event

You want $P(\text{at least 80 heads in 100 fair flips})$ by Monte Carlo. The true value is about $5.6 \times 10^{-10}$, so plain simulation almost never sees the event. Instead you simulate with a coin that lands heads with probability $0.8$.

(a) What weight must each simulated sequence carry so the estimator is unbiased? Write it in terms of the number of heads $H$.

(b) Why does this work, and why is $0.8$ a sensible choice for the biased coin?

(c) Roughly how many samples does each method need for a 1% relative error?