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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:

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

  2. 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.

  3. 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?

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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.

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  • $\begingroup$ That's awesome. While reading your answer, I also resolve something that was in the back of mind (when you said something like the preceding view of "reality") - and that is the following: a variable that is difficult to account for is that we live in a world where it is likely at some point that if they don't accept credit cards now - its certainly possible (perhaps likely) that they will in the future. It isn't like - 'well they didn't accept whale testicles yesterday - do you think they will today?' $\endgroup$
    – dgo
    Commented May 26, 2015 at 15:36

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