(I wasn’t able to find this specific question, so sorry if it’s been asked before)
Earlier today, I was told I would get a call between 5:00 and 6:00. If we make the assumption that this call is, for certain, going to occur within the specified time frame, I was under the assumption that every minute that passed would result in an increase in probability of the call would happen. For example, I was thinking that for the first minute the chance I get the call are 1/60, while the second minute would be 1/59 (as there are now only 59 minutes in which it could happen) and so on until the last minute is 1/1 (if I had still not received the call).
However, I was thinking back to my Freshman statistics course where my professor was talking about how the chances of the event happening at any given moment are the same, regardless of what time it is. In this case, at any given minute, the chance of me getting a call would still be 1/60. Unfortunately, I don’t remember any of the terminology that goes with these concepts, so I can’t remember if that actually applies to this problem.
So, in short, does the probability of something presumably inevitable increase as time passes? Or is the chance the same at any given moment?
In: Mathematics
It’s 5pm. The call will come in the next hour.
* What’s the chance it comes at 5:42? It’s 1 in 60.
* What’s the chance it comes at 5:03? It’s 1 in 60.
* What’s the chance it comes at 5:59? It’s 1 in 60.
It’s 5:15pm, you walk in on your friend, who you knew was waiting for the call. You can’t tell from their expression or body language whether the call has already come.
* What’s the chance it comes at 5:42? It’s 1 in 60.
* What’s the chance it came at 5:03? It’s 1 in 60.
* What’s the chance it comes at 5:59? It’s 1 in 60.
Now you ask them “did the call come?” Whatever they answer, you can now recalculate the probabilities – the *conditional probabilities*, conditioned on their answer.
Maybe they answer “yes, the call came.”
* What’s the chance it comes at 5:42? It’s 0. The call already came.
* What’s the chance it came at 5:03? It’s 1 in 15.
* What’s the chance it comes at 5:59? It’s 0. The call already came.
These are the *conditional probabilites* of the call coming in at different times, *given that* the call came before 5:15. You can calculate these with Bayes’ rule.
Or maybe they said yes. Then you want the conditional probabilites of the call coming in at different times, *given that the call did not come* before 5:15. Again, you can calculate these with Bayes’ rule.
* What’s the chance it comes at 5:42? It’s 1 in 45.
* What’s the chance it came at 5:03? It’s 0. The call didn’t come yet.
* What’s the chance it comes at 5:59? It’s 1 in 45.
*Conditional probabilities* are new probabilities that take into account new information, and can be calculated from the *a priori probabilities* using Bayes’ rule.
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