5: Probability
Probability describes the chance an event occurs over the long run
\[
P(A)\in[0,1]\\\
P(A\mid B)=P(A)+P(B)\\\
P(!A)=1-P(A)
\]
Law of large numbers: As the number of observations increases, the proportion of the occurrence approaches \(P(A)\)
Conditional Probability
Definitions
Or Condition
Probability of A or B occurring
If independent:
\[
P(A\cup B)=P(A)+P(B)
\]
And Condition
Probability of A and B occurring
If independent:
\[
P(A \cap B)=P(A) \times P(B)
\]
Given Condition
Probably of A given B has occurred.
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]