Let's see:

Case 1: Both show same
Case 2: Mis-match -- one heads, one tails
Case 3: One disappears in a burst of existential uncertainty
Case 4: Both disappear in a burst of existential uncertainty

So ... yeah ... 1/4.

:-)

(For my next magical trick I prove how reducing the temperature of an object to negative degrees Kelvin creates a container of negative volume into which you can stuff an infinite amount of stuff.)

((There.  That outta explode Mig's head.))

Unanswerable, without knowledge wether the coins are identical, and if not odds that two random coins will have the same persons head on the side.

Any idiot can face a crisis - it's day to day living that wears you out.

Maybe they mean "two heads"?

Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith

Are they tossed starting with both having the same side facing up? As shown in this paper, the probability of a coin ending up with the same side facing up as you started with is about .51, leading to a probability of about .5002 for your problem. (If the first author of this paper is the one flipping the coin, the probability is apparently whatever he wants it to be - his first career was as a professional magician...)