This has stumped me.

Assuming a fair coin.

2 flips the P(<=2 H)=1

3 flips the P(<=2 H)=0.875

4 flips the P(<=2 H)=.6875

whats the greatest number of times I can flip a coin and still have P(<=2 H) above 0.2?

Solution thoughts

Break down the probability distribution for 1 to x flips until 0.2 is obtained?

Assume that the data represents a part of a cumulative distribution and find the differrences between the values above sum to one and solve for the unknowns?