How do correlation and causality effect this scenario? I came up with a real world problem that I don't need to solve, but was intrigued by nonetheless.
Imagine I was trying to figure out if a supermarket chain took credit as well as cash.  My wife had been to the market yesterday and they didn't take credit.  That leads both of us to believe they won't take credit today (seems like a safe assumption).
Assume the following:


*

*The supermarket has not announced that they will accept credit soon - nor have they said they wouldn't.

*For this question - let's say it takes no real time  to implement a credit card system - they could implement it at the drop of a hat.

*We have no other reasons to think they will or will not decide to do this.

The fact that they didn't accept credit yesterday seems to increase the likelihood that they won't today (if we had been there last 1 year ago - we would be less likely to assume that they still didn't take credit cards).  But the fact that she was there yesterday is completely irrelevant to the stores credit card plans.  If they ever choose to implement credit card processing then it will have to happen one day - and that day could be any day - but it still seems we are grounded in our assumption that they won't accept them today.
My question is:
Does the fact that we knew that they didn't accept credit cards yesterday have any statistical correlation to whether they will accept them today?  And if so; why?
 A: If we know that the decision whether to accept credit or not on a particular day
is made that morning by flipping a coin 
("Heads we accept credit, tails we don't"),
then of course there is no correlation between the outcomes of your wife's
attempt to use credit yesterday and your attempt to use credit today.
In reality, it is highly unlikely for such a decision to be made that way.
A store would make a decision about accepting credit or not based on
how its management thinks that decision would support its business plan.
Having made that decision once, management might reconsider it if new
information came along, or they might just decide to reassess the
decision periodically, but they are very unlikely to change policy one day
and change it back the next.  Hence any policy tends to persist for
several days (at least). The policies on consecutive days will be
correlated, because if you pick any random pair of consecutive days,
the chance that they fall within one of the periods of no-change-in-policy
is higher than that they happen to fall exactly across one of the
boundaries between policies.
I think it is hard for someone to approach a question like this without
applying something like the preceding view of "reality" to the question,
at least if they live in a real-world society
that might form the setting for that question.
Hence your intuition that there will be a correlation between the
store's credit policy yesterday and its policy today.
A visitor from another planet with no knowledge of human culture might not
intuit such a correlation. They might not even assume that the refusal to
take your wife's credit card yesterday was a policy against taking any
credit cards that day; perhaps the decision is made randomly and
independently for every single customer.
The extraterrestrial visitor might, however,
apply some kind of Bayesian inference to the situation, in which
there was a prior probability that the store "will accept a credit card",
and after your wife is unable to use her card, the posterior probability
of "will accept a credit card" is lower than the prior probability.
The posterior probability is what applies to your use of a credit card,
hence the probability that your card will be accepted is lower than
it would have been if we did not know that
your wife's card had not been accepted.
Note, however, that this Bayesian inference by the extraterrestrial
is not a statistical correlation,
at least not in the usual (frequentist) sense.

