Assume the probability of flipping a coin to heads 20 times in a row is 1:1,000,000

Q: Each day you wake up and flip a coin exactly 20 times. What are the odds that on one of these mornings you flip exactly 20 heads in a row?

A: I believe the answer is 1:20,000,000

The odds increase due to the 20 consecutive (Heads) having to perfectly correspond with the 20 flips on any given morning. Which would be 1:20.

Is this correct?

Now this is where my brain starts hurting.

Q: Assume that answer is correct, and I could attain such an age. What is the probability that after 20 Million mornings I would have accomplished this feat at least once?

A: I am at a loss. It seems like the answer is somewhere in between "very likely" and 100%. How does one arrive at a more precise number?

Thanks in advance to anyone who reads this.