What are the allowed operations? Can we only use the standard operations (connect two points or extend a line segment) or may we use other tricks, for example align the straightedge along the diameter and use the other side of the straightedge to draw a parallel line?
1. Nobody needs to sacrifice himself. The first prisoner to guess has a 50/50 chance of survival, regardless of whether he adheres to the strategy or not. Nothing he can do can improve his chances of survival, so he might as well help the others (not assuming any grudges).
2. Well, yes, the average prisoner most likely is more stupid than the average non-prisoner. But this being a math/logic puzzle, aimed at people who are more intelligent than average, the general assumption is that the prisoners would be intelligent enough to understand the solution proposed by the puzzle solver. And anyway, in this fictional setting with a king who likes to toy with his people, it is plausible that these prisoners have survived through similar challenges before and thus are far more intelligent than regular prisoners, through the survival of the fittest.
3. I could apply the same argument as for point 2, that this is a math/logic puzzle and we should not assume distrust unless explicity told to do so. And I could also argue that in this fictional world with an oppressive king, the people might have developed more trust towards each other.
4. The OP's description of the procedure seems to imply that each prisoner's judgment is carried out before they move on to the next prisoner. So unless you have heard a prisoner die before you (except the first one who might die due to random chance), you would have no reason to assume that a mistake has been made. And again, if we were supposed to account for the possibility of mistakes, we would have been explicitly told to do so, and possibly even have been provided with the exact probability of a mistake being made.
5. Finally, even with all that being said, all it takes for one mistake/betrayal to be canceled out is another mistake/betrayal. Maybe the first prisoner intended to betray you, but was too stupid and accidentally gave the "good" answer. So even if the prisoners were indeed selfish, stupid, distrusting, and prone to betrayal, it would boil down to a crap shoot anyway, even if you did follow the strategy. So once the strategy has been proposed, there is really no reason to deviate from it (unless you have heard people dying before you as I mentioned earlier).
How do you arrive at this answer? If I'm not mistaken, the answer is approximated by running the following program indefinitely, given a full list of primes in order P1 = 2, P2 = 3, P3 = 5, and so on:
1: set S = 0, i = 1
2: set S = S + (1-S)/Pi2
3: set i = i+1
4: return to step 2
As this program keeps going, S should tend to the desired probability. Not being a programmer, I haven't been able to run this for large numbers, but it seems to me that it would converge closer to 0.4 than 0.6. I would be very surprised if it converged at exactly 6/pi2, as this would indicate a clear correlation between pi and the set of primes.