Independent guessing gives the prisoners 7.9 x 10^-31. Following the slip you just found gives them 0.311828, and the gap is thirty orders of magnitude from a rule you can state in one sentence. The strategy never raises anyone's individual chance above one half; it only makes the failures coincide, which is the whole lesson.
Also asked as100 prisoners · hundred prisoners problem · locker puzzle · prisoners and boxes
The midpoint of p and q sits strictly between them, and consecutive means precisely that no prime lives in that interval, so the answer is never and the proof is two lines with no arithmetic in it. The pair 2 and 3 survives for a different reason, since five halves is not an integer, and it is the only such pair.
Multiplying by the conjugate turns the difference into x over the sum of the same two terms, exactly, with no series and no error term, and dividing through by x reads off the limit one half. Completing the square gives a constant excess of a quarter, so the gap climbs toward a half strictly from below and never reaches it.
The factor pi over four cancels off both sides, so the comparison is 324 against 244, about a third more pizza. Written that way it is the law of cosines: lay the three widths out as a triangle and the corner between the two smaller sides opens to 109.47 degrees, wider than square, which is the answer with no arithmetic at all.