Fringe was only recently renewed, but the debate has already begun about whether Season 4 should be the last. Some argue the odds of a fifth season (while not impossible) are so slim that the writers should take the opportunity to end the series well, rather than risk leaving viewers hanging. Others counter that, because the series could conceivably continue, writing S4 is if it were the last might leave the writers with no story to tell in S5.
Both arguments have merit, so I decided to compare the Expected Value of the two aproaches to see if we can resolve this objectively. For those unfamiliar, Expected Value is simply a weighted average of all possible outcomes. For example, the probability of a six-sided die landing on any one side is 1/6 or about .167. The Expected Value of a die roll is thus 3.5 because .167(1)+.167(2)+.167(3)+.167(4)
So, what does this have to do with Fringe? Let's assume S4 and S5 could each rate a maximum of 10 points in quality. Let's further stipulate that the probability of the show's being renewed for S5 is around 35%, a rough but reasonable estimate offered by one proponent of the second approach (i.e., that the writers should hedge their bets on S5). By comparing the Expected Value of each approach, we can get a sense of which course is preferable.
Approach #1: The writers write S4 as if it's the last. There's a 65% chance they're right, in which case S4 scores a perfect 10 in quality. But there's also a 35% chance they get renewed after blowing their load in S4. In that case, S4 is still great but S5 suffers and rates only a 5 in quality. The Expected Value of this first approach is thus 11.75 because .65(10)+.35(15)=11.75.
Approach #2: The writers write S4 with an eye towards renewal. There's a 65% chance they're wrong, in which case S4 rates only a 5 in quality because they leave viewers hanging. But there's also a 35% chance they're right, in which case both S4 and S5 rate perfect 10s in quality. The Expected Value of this second approach is thus 10.25 because .65(5)+.35(20)=10.25.
From a probabalistic perspective, Approach #1 is objectively superior to Approach #2 because it maxmizes expected quality despite the possibility the writers might be caught with their pants down in S5. In fact, the same is true even if you assume that the probability of renewal is more like 49%, though it's a closer call. No matter how you slice it, therefore, the writers would be wise to write S4 under the assumption it's the last.