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ece108: add conditional probability

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eggy 2023-03-29 18:43:29 -04:00
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@ -911,3 +911,19 @@ $$Pr\{X^{-1}(\{x\})\}$$
Thus the **binomial distribution** for $r$ successes of $n$ total tries, if they are independent, is: Thus the **binomial distribution** for $r$ successes of $n$ total tries, if they are independent, is:
$$Pr\{X=r\}{n\choose r}p^rq^{n-r}$$ $$Pr\{X=r\}{n\choose r}p^rq^{n-r}$$
### Independence
Please see [SL Math - Analysis and Approaches 2#Conditional probability](/g11/mcv4u7/#conditional-probability) for more information.
Two events are independent if they can be treated separately.
$$\text{independent}\iff Pr\{A\cap B\}=Pr\{A\}Pr\{B\}$$
Or, via the inclusion-exclusion theorem:
$$\text{independent}\iff Pr\{A\cup B\}=Pr\{A\}+Pr\{B\}-Pr\{A\}Pr\{B\}$$
**Bayes' theorem** provides a general formula for conditional probability:
$$Pr\{A|B\}=\frac{Pr\{B|A\}}{Pr\{B\}}$$