I do look at these probabilities before the season starts, so I can see which teams are likely to do well over the course of the long season. Half way through the season, I will look at them to see who should buy or sell. I just can't myself to care about them over a short period. At this point, if your team is alive they are alive. Probabilities are always in my head somewhere, but when I watch the game tomorrow I am not going to care whether Fangraphs thinks the Tigers have a 40% probability or a 60% probability. They were less than 10% just a few games ago.
On the other hand, I think I get why it might be fun to watch them go from 10% to 100%.