Odds of guessing my way to $1M--the answer

For those who have an understanding of probability, I hope you have taken a stab at this yourself--check out the "Assumptions" post before reading this if you want to try to solve this on your own.

Let's start with the simplified question that I pose at the end of that post:

Simplified Question:
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?

Solution to Simplified Question:
In this case, in order to win $1M, I need to first guess the first question correctly (probability = 1/4), and then also guess the second question correctly (probability = 1/4).  Since these are independent events, the probability of them both happening is the product of the two individual probabilities, or

1/4 * 1/4 = 1/16.

Alternatively, you could calculate my odds of NOT winning $1M using this strategy.  In this case the math goes as follows:

3/4  + 1/4 * (3/4)

(first term represents 3/4 of time I miss the first question.  Second term says of the 1/4 times I get the first question right, I will then miss the second question 3/4 of the time.)

= 3/4 + 3/16 = 12/16 + 3/16 = 15/16.

This agrees with the first result.  I have 1/16 chance to guess right, and 15/16 chance to fail.

***

Original Question:
Now, on to the original question...what are my chances to guess right on all 14 questions at the start of the game?  As I mention on the air, it is important to factor in the 50/50, so the odds I state on the air (rounded) are the odds of guessing 13 questions where I have 1/4 chance on each, and one question where I have a 1/2 chance.

Last chance to work it out on your own!  Look under pic for the solution.

Okay, the same logic applies as with the simplified question...we multiply the probabilities of guessing each of the individual questions right, so this works out to:

(1/4)^13 * (1/2)

(the "^13" there means 'raised to the 13th power')

= (1/2)^26 * (1/2) = (1/2)^27

Or, one over 2 raised to the 27th power.

or 1/134,217,728 (I rounded to 'about one in 135 million' on the show.)

As for the vending machine statistic...I did not cite my source, but it is from the internet (so it must be true!)  I saw it referenced in several places, though I suspect there is some original source somewhere, and then others keep referencing that.  Here is one of the pages I found:

Lottery and crushed by vending machine stats