# Chain Rule for Probability

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## Theorem

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr B > 0$.

The **conditional probability of $A$ given $B$** is:

- $\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$

## Proof

The validity of the material on this page is questionable.In particular: This is not a mathematical proof. There is no rigor at all, isn't it? Kind of a justification of the definition $\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$ comparing to the real world. But maybe an ancient mathematics, surely not the modern mathematics.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Suppose it is given that $B$ has occurred.

Then the probability of $A$ having occurred may not be $\map \Pr A$ after all.

In fact, we *can* say that $A$ has occurred if and only if $A \cap B$ has occurred.

So, if we *know* that $B$ has occurred, the conditional probability of $A$ given $B$ is $\map \Pr {A \cap B}$.

It follows then, that:

- if we
*don't*actually know whether $B$ has occurred or not - but we know its probability $\map \Pr B$

we can say that:

- The probability that $A$ and $B$ have both occurred is the conditional probability of $A$ given $B$ multiplied by the probability that $B$ has occurred.

Hence:

- $\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$

$\blacksquare$

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**conditional probability**

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- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.6$: Conditional probabilities: $(19)$ - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation