Here's the question.
You are flying to Zanzibar (Tanzania) with a connection in Aachen (Germany). You checked a bag. Assumptions:
- There's a 50% chance that your bag successfully made the connection in Aachen. Conversely, there's a 50% chance that your bag is stuck in Aachen.
- The baggage crew in Zanzibar loads all the bags onto the carousel in 10 min, at a constant rate.
You're sitting in Zanzibar airport waiting for your baggage. You can deduce the probability that your bag is stuck in Aachen. At t=0min, the chance that your bag is stuck in Aachen is 50%. At t=10min, if all bags have been loaded and your bag is not there, congratulations, your bag is 100% stuck in Aachen.
But how about at t=5min? If your bag is still not on the carousel in Zanzibar, what's the chance that your bag is stuck in Aachen? I thought it was 75%, but the book says it's 66%! I can't figure out why.
Edit1: my rationale for saying it's 75% is since the rate of baggage-loading in Zanzibar is linear, and the percentage goes from 50% to 100% in 10 min, the midpoint probability at t=5 would also be at midpoint: 75%.
Edit2: As suggested in the comments, I'm framing this in Bayesian terms. $$ P(stuck \mid t_5) = \frac{P(t_5 \mid stuck)P(stuck)}{P(t_5)} $$ Where $t_5$ denotes the state where there's no bag on the carousel at 5min. Therefore, it'll be, $$ \frac{1 * 0.5}{0.75} = 0.666 $$ Thanks! That solved it. Appreciate it, Demetri.
