But then I thought, suppose

-there is a snag in customs?

-something else happened?

-she is somehow not on this flight at all?

so the probability of her NOT showing up any minute was also increasing.

How can both be true at the same time?

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But then I thought, suppose

-there is a snag in customs?

-something else happened?

-she is somehow not on this flight at all?

so the probability of her NOT showing up any minute was also increasing.

How can both be true at the same time?

"Given your wife has not shown up before time \( t \) and all information up to time \( t \), the probability of your wife showing up within the future time interval \( [t, t + \Delta t] \)"

It seems that at the very beginning, you have a strong belief that every thing is alright and expecting she will show up, and that's why you claim that the probability is increasing.

Maybe this can be used to illustrate the point:

Denote \( X \) be her show up time. \( E \) be the event that she has taken the current flight.

Then the required probability is

\( \Pr\{X \in [t, t + \Delta t]|\mathcal{F}_t\} \)

\( = \Pr\{X \in [t, t + \Delta t]|\mathcal{F}_t\}|E, \mathcal{F}_t\}\Pr\{E|\mathcal{F}_t\} + \Pr\{X \in [t, t + \Delta t]|\mathcal{F}_t\}|E^c, \mathcal{F}_t\}\Pr\{E^c|\mathcal{F}_t\} \)

by law of total probability. The first term is somehow increasing over time at the beginning, but the "weights" \( \Pr\{E|\mathcal{F}_t\}, \Pr\{E^c|\mathcal{F}_t\} \) actually shifting towards the later over time so it is decreasing later.

Hopefully not misunderstood your problem

This has to be a commom problem in some field, say engineering!