Odds of guessing my way to $1M--assumptions

If you have seen my run on Who Wants to be a Millionaire (aired on 1/29/2019 and 1/30/2019, available to stream here: Rich Day 1: Aired 1/29/2019 , Rich Day 2, Aired 1/30/2019 ) then you know that early on Day 1, Chris Harrison and I talk about a probability math question related to the show.  Specifically, what are the odds that I would answer all 14 questions correctly by sheer guesses, assuming that I have no idea what the answer is on any question?  I immediately state the rounded answer (1 in 135 million), and then make the vending machine joke.  Typical source--you'll find more if you search  Believe it or not, I think that this little exchange is what helped me pass the interview to get on the show!

I'm hoping people will "check the math", and try to corroborate (or disprove) my answer.  There are some important assumptions in my work:

First, as stated in the show, I assume that I have absolutely no clue on any of the 14 questions.  This is admittedly a rather ridiculous assumption, given the intentionally easy questions they pose in the earlier rounds, but we are assuming the worst case scenario here, so for the purposes of this problem, I will need to guess on every single question.

Next, I will certainly take advantage of my three lifelines, but I am again assuming the worst case:

• I will use my "ask the audience" lifeline, but for the purposes of this question, we will assume that the audience also has no clue, and that this lifeline does not help me out at all (assume after I ask the audience, each answer gets 25% of the vote).  Ugh.
• I will use my "plus-1" lifeline, but sadly for the purposes of this "worst case" scenario for the problem, he too is unable to help me in any way.  Strike two.
• I will use my "50/50" lifeline at some point, where two of the wrong answers are eliminated, and I only have to guess between the other two.  Yes!!  This will definitely help me, as my chances of getting that question right are 1/2, rather than the 1/4 chance I will have on the other 13 questions.  

Okay, now that we are on the same page with the assumptions, you can check my math--what are my chances of guessing my way to a million bucks?

If your probability is rusty, consider this easier question, which might get you on the right track:


Imagine that somehow I have answered 12 of the 14 questions correctly, using all of my lifelines along the way.  So in this fantasy scenario, I only have 2 more questions to go to win the $1,000,000.  What are my odds of winning $1,000,000 at that point, if my strategy is to just guess on the final two questions?