Hi, long time lurker. Great writing on both GM's and player's part, thank you everyone.
Also you've made me get my statistics course notes from their dusty resting place and make an account to share their revelations with you on this specific sealing failure scenario. Although now that I think on it, maybe this math has already been done elsewhere previously? In which case disregard this message except for the gratefulness I feel to you all.
Onto the math: Let's call the probability of one of these seals failing in a spectacular fashion when used "poootsfiasf" or "p" for short. Then the probability "P" of there being no failures when there have been "N" activations is:
P=(1-p)^N
So let's use this to try and make a guess for the failure probability. Assuming that the seals that have been used since their creation with no ill effects, and that they have been used about 10 times per year since their creation 20 years ago (numbers that I am pulling from thin air), this gives us 200 uses. Now, how lucky do we think Hidden Mist has been all these years? Let's say the cumulative probability that none of these 200 seals has malfunctioned is 99% (again, thin air, could choose anything between 0.00...01 and 99.99...%). So we plug this into the equation and have:
0.99=(1-p)^200
p=1-200nth_root_of(0.99)=0.00005 or thereabouts.
Further assuming that the quality of the seals used on the exam is the same as those that have been used in the past (because sure working nonstop is great for not making mistakes and who's counting assumptions anymore anyway):
P=(1-0.00005)^4000=0.8
So a 20% chance that there is trouble.
A nice formula for the relationship is also:
P1^N2=P2^N1
P1=P2^(N1/N2)
Which relates the probabilities of nothing happening on two different accounts (P1, P2), and the number of times fate was tempted in each case (N1, N2), assuming the poootsfiasf factor stays the same.
Here are some test values. Remember, P1 is the probability that everything will be fine *heh*.
N1 N2 P2 P1
4000 200 0.99 0.82
3000 200 0.99 0.86
4000 400 0.99 0.90
4000 200 0.9 0.12
4000 100 0.99 0.67
4000 200 0.999 0.98
As we can see, the biggest factor is how close to sealing failure Mist has been in the past
Best Regards.