Consider the following joint distribution for the random variables A and B:

\begin{array}

{|r|r|}\hline & B=1 & B=2 \\

\hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\% \\

\hline

\end{array}Intuitively,

- if I know A, I can predict very well B (98% accuracy!)
- but I if know B, I can’t say anything about A
Questions:

- can we say that A causes B?
- if yes, what is the mathematical way to conclude that A causes B?
thank you! (and apologies for the maybe “naive” question)

**Answer**

can we say that A causes B?

No, this is (presumably) a simple observational study. To infer causation it is necessary (but not necessarily sufficient) to conduct an experiment or a controlled trial.

Just because you are able to make good predictions does not say anything about causality. If I observe the number of people who carry cigarette lighters, this will predict the number of people who have a cancer diagnosis, but it doesn’t mean that carrying a lighter causes cancer.

Edit: To address one of the points in the comments:

But now I wonder: can there ever be causation without correlation?

Yes. This can happen in a number of ways. One of the easiest to demonstrate is where the causal relation is not linear. For example:

```
> X <- 1:20
> Y <- 21*X - X^2
> cor(X,Y)
[1] 0
```

Clearly `Y`

is caused by `X`

, yet the correlation is zero.

**Attribution***Source : Link , Question Author : elemolotiv , Answer Author : Robert Long*