# An event and its complement

#### Outlier

##### TS Contributor
My wife was overdue clearing customs so I thought the probability of her showing up any minute was increasing.

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?

#### BGM

##### TS Contributor
It seems that you need to define the event more precisely, say

"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.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Yes, it needed conditional terms. What does the following represent?

$${F}_t$$

I love this question and would be interested to see if it could be graphically displayed!

#### Outlier

##### TS Contributor
Well, I'm not so embarrassed now that I couldn't figure this out. This problem has at least one other problem inside it.

The answer is I shouldn't hang around airports! :yup:

#### Outlier

##### TS Contributor
I cut and pasted your answer and will review my Baye's material. Thanks!

#### BGM

##### TS Contributor
The $$\mathcal{F}_t$$ is a standard notation for the filtration up to time $$t$$ - representing the information observed in the past.

I just try to make the description plausible. Hopefully it at least help a little bit.

P.S. Was sad when TS was down a few days.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Have I said that I like this problem, because I do. Definitely Bayesian aspects, however I was just thinking that time has some type of threshold and flips so it is like an autoprogressive meets quadratic ("V" shaped) or say inverse velocity of a ball thrown in an arc or straight-up, well not exacttly. At some tipping point it flips. Could also be tackle with a couple of receiver operator curves perhaps.

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