math calculation for Who Wants To Be A Millionaire? TV Gameshow

What’s the correct math calculation for deciding whether or not to hazard a guess on the next Millionaire question?

The payouts for the initial round are as follows: $25,000, $15,000, $10,000 …

The contestant starts before the first question with a bank of zero, so a complete guess (4 choices translates to 25% of randomly guessing the correct answer) is always better than walking away before the first question (besides the fact that you can skip the first two questions if nothing else).

After the first question, the contestant’s average bankroll will be ($25,000 plus $15,000 + …) divided by the total number of initial round payouts. Without knowing the precise numbers, my guess is about $7,000.

That would leave about $6,000 average for the next question’s payout, so the total estimated monetary outcome of a correct answer for question #2 would be about $13,000.

A random quess with 25% chance would yield $13,000x0.25= about $4,250
versus $1,000 x0.75 = 750 for a wrong answer, so it pays to randomly guess on question #2.

But although $6,000 would be the average payout in general, for any particular contestant, it would be calculated by subtracting the actual dollar amount of the first round then doing the averaging.

Then factor in that a contestant may be 99.9% certain that a particular choice is wrong although unsure of the other choices. That would change the percentage for a random correct guess to 33.33%, and if the contestant is certain that two choices are wrong, then the random guess becomes a 50/50 outcome.

Has anyone calculated the math on when to proceed on a random guess based on payout odds?

To rephrase the question, how do you calculate the expected value of randomly guessing the next Millionaire question (taking into account knowledge of any incorrect answers, the fact that the contestant’s bankroll is cut in half if (s)he walks away, etc.)?

For question #2, the EV comparison would be
$7,000/2=$3500 versus ($1000x.75)+($13000x.25)=$5000
so a random guess would have a higher EV than walking away.

For question #3, the EV comparison would be
$13000/2=$7500 versus ($1000x.75)+($18000x.25)=$750+$4500=$5250
or if one answer is certainly wrong then
$7500 versus ($1000x.67)+($18000x.33)=$6670
or if a tossup between two answers then
$7500 versus $9500

If contestant’s game bankroll is less than $10,000 by question #3, they should continue even with a completely random guess (unless their personal bankroll is poverty level and $5000 is money they can’t afford to lose).