Formula Sheet:

Probability of an event \(E\):

\[P(E)=\frac{\text{number of outcomes in the event }E}{\text{number of outcomes in the sample space}}=\frac{|E|}{|S|}\]

Probability of the Complement of an event \(E\):

\[P(\overline{E})=1-P(E)\]

Empirical Probability:

\[P(E)=\frac{\text{frequency of the class}}{\text{total frequencies in the table}}=\frac{f}{n}\]

Addition Rule (general):

\[P(A\text{ or } B)=P(A)+P(B)-P(A\text{ and } B)\]

Addition Rule (mutually exclusive events):

\[P(A\text{ or } B)=P(A)+P(B)\]

Multiplication Rule (Independent Events):

\[P(A\text{ and }B)=P(A)\cdot P(B)\]

Multiplication Rule (dependent Events):

\[P(A\text{ and }B)=P(A)\cdot P(B)\]

The arrangement of \(n\) objects in a specific order using $r$ objects at a time (permutations)

\[_nP_r=\frac{n!}{(n-r)!}\]

The number of permutations of \(n\) objects when \(r_1\) objects are identical, \(r_2\) objects are identical,...,\(r_p\) objects are identical, etc. is: (where \(r_1+r_2+...+r_p=n\))

\[\frac{n!}{r_1!r_2!...r_p!}\]

The number of \(r\) objects selected from \(n\) objects is denoted by \(_nC_r\) and is given by the formula (combinations)

\[_nC_r=\frac{n!}{(n-r)!\cdot r!}\]