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Example:

• Dentist: “You have multiple cavities.”




• Patient: “That’s ridiculous! You always told me that brushing my teeth prevents cavities. I brush my teeth every night. Therefore, I can’t possibly have cavities.”




• Dentist: “It is true that brushing your teeth has been proven to help prevent cavities. However, in your particular case, there could be other factors negating the positive effects of brushing your teeth, e.g. you aren’t brushing long or thoroughly enough.”

In the above scenario, the patient thinks that simply because they brush their teeth, they are guaranteed not to get a cavity, i.e. the patient thinks that there is a simple, causal relationship between brushing teeth and not getting cavities. In reality, only brushing one’s teeth thoroughly and completely, for a sufficient amount of time, with a proper brush, etc. will prevent cavities. In other words, there are multiple factors at play that determine whether brushing one’s teeth will produce the desired effect or not.

What do you call the fallacy of thinking that if A statistically causes B, then A implies B?

For the original title quoted above, the closest is probably correlation implies causation, deducing a cause-and-effect relationship solely on the basis of an observed statistical correlation. The Latin name is cum hoc ergo propter hoc (with this, therefore because of this).

A remark on the terminology. First, "statistically causes" is not an expression used, if A causes B then it is not merely statistical. If it is, or causation is unknown, people say that "A correlates with B", or that "B tracks A". Second, "implies" does not distinguish between correlation and causation, "A implies B" can be true even if A has nothing to do with causing B. For example, "John is Mary's brother" implies "Mary is John's sister", but neither causes the other.

However, I am afraid, the OP example does not really match the (intended) title. The problem is not that brushing teeth is not causally related to not having cavities. It is. The problem is rather that confounding factors (such as quality of brushing, gum condition, etc.) are overlooked. So this is closer to the fallacy of the single cause, "a single, simple cause of an outcome when in reality it may have been caused by a number of only jointly sufficient causes", or spurious correlation, "two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor".

The patient has confused necessary and sufficient conditions. Brushing one's teeth is necessary for good dental health, but is not sufficient to guarantee that outcome.

Your example is not well-suited to your question, or vice versa. In your example, the patient (as @Barmar notes) has misunderstood the claims about brushing one's teeth. A better example might be (grimly) lung cancer, where the patient's line is "(A)Smoking causes lung cancer, (B)I don't smoke, therefore (C)I can't have lung cancer". Obviously, it's possible that other things can cause lung cancer, so while A and B may be true, the implicit assumption that ~(A & B) -> C is incorrect.

What do you call the fallacy of thinking that some action A will guarantee some outcome B, when in reality B depends on multiple other conditions?

I agree with Conifold, possibly with a better explanation.

Faulty relevance logic, it suffers in many respects and lacks critical thinking and rationality. A false consensus effect causes them to question the statement made by the dentist, probably the reason they went to see the dentist in the first place; and one would hope a mistake that the dentist (or even a layperson) would not make.

If the patient didn't accept the explanation offered it becomes conservatism, it's a confirmation bias if they were told that if they brush their teeth they won't get any cavities; an anchoring and belief bias.

The availability cascade of having brushed their teeth previously and never had cavities, and continuing to have brushed them, leads them to the belief that they couldn't have cavities - not sure why they thought the need to visit an expert and then call into question a simple truth.

There's a list of cognitive biases but it falls closest to the fallacy of the single cause:

"The fallacy of the single cause, also known as complex cause, causal oversimplification, causal reductionism, and reduction fallacy, is a fallacy of questionable cause that occurs when it is assumed that there is a single, simple cause of an outcome when in reality it may have been caused by a number of only jointly sufficient causes.

It can be logically reduced to: " X caused Y; therefore, X was the only cause of Y" (although A,B,C...etc. also contributed to Y.)

Causal oversimplification is a specific kind of false dilemma where conjoint possibilities are ignored. In other words, the possible causes are assumed to be "A or B or C" when "A and B and C" or "A and B and not C" (etc.) are not taken into consideration.".

In boolean logic, there's something called the Contraposition
https://en.wikipedia.org/wiki/Contraposition

If you consider the statement "if X then Y" (X -> Y),
there exists some opposite of the statement.

One common flaw while inverting conditional statements is not correctly calculating the contraposition.

The mistake is introduced inverting the left and right side of the argument

"if NOT X then NOT Y" ('X -> 'Y)

This is incorrect. If you read the above wikipedia entry, the contrapositive is

"if NOT Y then NOT X" ('Y -> 'X)

Your example is not quite clear.

Patient: “That’s ridiculous! You always told me that brushing my teeth prevents cavities. I brush my teeth every night. Therefore, I can’t possibly have cavities.”

If we take "X prevents Y" to mean "if X, then not Y", and if the dentist did in fact say that brushing prevents cavities, and the patient did in fact brush their teeth, then the patient not getting cavities does in follow from the dentist's statement. The only room for this not being a logical conclusion is if the dentist in fact made a weaker claim, such as "brushing reduces the incidence of cavities" and the patient misremembers that, or if we take "prevent" to mean merely reduce the incidence. Neither is really a fallacy; the first is a failure of memory, that may have cognitive biases as an influence, while the second is a miscommunication due to ambiguity. If the patient's conclusion was based on false beliefs, such as what terms the dentist used or what intent the dentist had in using those terms, rather than faulty logic, then the argument is unsound, but not fallacious.