- If each event isequally likely to happen, then one usually assigns a probability
valueP(ei) =^1 nto each event as then
∑n
i=1P(ei) = 1.- The probability assigned to the entire sample space is unity and one writes
P(S) = 1.
Probability of an Event
After assigning probabilities to each simple event of S, it is then possible todetermine the probability of any eventE associated with events fromS. Consider
the following cases.
- The eventE=∅is the empty set.
- The eventEis one of the simple eventseifromS(ifixed, with 1 ≤i≤n)
- The eventEis the union of two or more events fromS.
For the case 1, define the probability of the empty set∅as zero and writeP(∅) = 0.
In case 2, the probability of eventE is the same as the probabilityP(ei) so that,
P(E) =P(ei). Consider now the case 3. IfEis an event associated with the sample
spaceSandEis its complement, then
P(E) = 1−P(E) (11.19)This is known as thecomplementation rulefor probabilities.
If E 1 andE 2 aremutually exclusive eventsassociated withS, thenE 1 ∩E 2 =∅and
P(E 1 ∪E 2 ) =P(E 1 ) +P(E 2 ), E 1 ∩E 2 =∅ (11.20)In general, ifE=E 1 ∪E 2 ∪···∪Emis an event andE 1 , E 2 ,... , Emare mutually exclusive
events associated withS, then the intersection givesE 1 ∩E 2 ∩···∩Em=∅and the
probability ofE is
P(E) =P(E 1 ∪E 2 ∪···∪Em) =P(E 1 ) +P(E 2 ) +···+P(Em) (11.21)This is known as theaddition rulefor mutually exclusive events.
If E 1 andE 2 are arbitrary events associated with a sample spaceS, and theseeventsare not mutually exclusive, then
P(E 1 ∪E 2 ) =P(E 1 ) +P(E 2 )−P(E 1 ∩E 2 ) (11.22)HereP(E 1 )is the sum of all the simple events definingE 1 andP(E 2 )is the sum of all
the simple events definingE 2. If the events are not mutually exclusive, then the sum