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math: add more probability rules and slight refactor

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eggy 2021-03-08 18:41:46 -05:00
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@ -174,11 +174,20 @@ $$P(A)=\frac{n(A)}{n(U)}$$
As event $A$ must be a subset of all possible outcomes $U$, where $1$ indicates that the event always happens and $0$ the opposite:
$$0\leq P(A)\leq 1$$
The complement of **event A** is the probability that it does not happen. It is written as $A^c$, $A'$, or $\pu{not } A$.
The **complement** of event $A$ is the probability that it does not happen. It is written as $A^c$, $A'$, or $\pu{not } A$.
$$P(A')=1-P(A)$$
Events $A$ and $B$ are disjoint if no outcomes between them are common and can never happen simultaneously. As such the probability of one of the events happening is equal to their sum.
$$P(A\cup B)=P(A)+P(B)$$
Events $A$ and $B$ are **disjoint** or mutually exclusive if no outcomes between them are common and can never happen simultaneously. As such the probability of one of the events happening is equal to their sum.
$$
P(A\cup B)=P(A)+P(B) \\
P(A\cap B)=0
$$
Events $A$ and $B$ are **exhaustive** if their union includes all possible outcomes in the sample space: $A\cup B=U$.
$$P(A\cup B)=1$$
The **principle of inclusion and exclusion** forms a general rule for the union between two sets:
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
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