\(P(\phi)\) is 0.
For any finite collection \(E_1,E_2,...E_m\) of disjoint event \[ P(E_1 \cup E_2 \cup .... \cup E_m)=P(E_1)+P(E_2)+...+P(E_m) \]
If \(A\subseteq B\), then \(P(A)\leq P(B)\)
\(P(A^c)=1-P(A)\)
$P(AB)=P(A)+P(B)-P(AB) $
\(P(A\cup B \cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)\)
The Multiplication Rule: If a task can be completed in K stages and stage i has n_i outcomes, regardless of the outcomes of the previous stages, then the task has \(n_1,n_2,..n_k\) outcomes.
Example. coin tossed 10 times.
\(2^{10}=1024\)
Select first second, and third place winder from a group of 4 final-list. (select without replacement.)
\(4\times 3\times 2=24\)
The probability of an event \(E\) is the likelihood of the occurrence of E, This is denoted \(P(E)\). (\(0\leq P(E) \leq 1\))
Let S be a sample space with N outcomes that are equally likely to occur. Then the probability of each outcome is 1/N.
If \(N(E)\) denotes the number of outcomes in the Event \(E\), then
\[ P(E)={{N(E)}\over N} \]
Example: Suppose we roll 2 die separately, the probability of rolling 6 is.
\(E=\{(1,5),(2,4),(3,3),(4,2),(5,1)\},N=5\)
\(P(E)=5/36=0.139\)
If a distinction is made between the outcomes of the stages, we say the outcome are ordered. Otherwise we say the outcomes are unordered.
The ordered outcomes are called permutations of k units. The number of permutations of k unites selected from a group of n units is denoted by \(P_{k,n}\).
The unorderd outcomes are called combinations of k units. The number of combinations of k units selected from a group of n units is denoted by \((^n_k)\).
To compute the number of permutations of k units:
\[ P_{k,n}=n(n-1)(n-2)...(n-k+1)={n!\over{(n-k)!}} \] where \(m!=m(m-1)(m-2)..(2)(1)\) and \(0!=1\).
and \(P_{n,n}=n!\)
To compute the number of combinations of k units
\[ \begin{pmatrix}n\\k\end{pmatrix}={P_{k,n}\over{P_{k,k}}}={n!\over{k!(n-k)!}} \] choose k form n
Example. Select two card from a deck of 52 cards.
\[ P_{2,52}={52!\over{(52-2)!}}=52\times 51=2652 \] * When both card to player 1
\[ C_{2,52}=\begin{pmatrix}52\\2\end{pmatrix}={{52\times 51}\over 2!}=1326 \]