Waiting at Tram Stop

You arrive at a tram stop and notice that the tram hasn’t arrived, even though it’s already scheduled time. The display calmly reads:

“The tram will arrive anytime within the next n minutes.”

At this moment, you form a belief: with high probability, the monitor is correct and the tram is just running late. But with a small probability e, you suspect something is off, perhaps the tram will not arrive at all.

As time passes and no tram appears, your belief begins to shift. The longer you wait, the more plausible the “never arriving” scenario becomes.

Now suppose you check the monitor repeatedly—say, every minute. Each time you see nothing, you accumulate negative evidence. Your belief that the tram will never come starts to increase.

At what point does your belief exceed 50% that the tram will never arrive? And how does the frequency of checking affect this threshold?