**9.3 Probability of a Combined Event**

**9.3a Finding the Probability of a Combined Event by Listing the Outcomes**

**1.**A combined event is an event resulting from the union or intersection of two or more events.

**2.**The union of combined event ‘

*A*or

*B*’ =

*A*υ

*B*

**3.**The intersection of combined event ‘

*A*and

*B*’ =

*A*∩

*B*

**Example:**

Diagram below shows five cards labelled with letters.

All these cards are put into a box. A two-letter code is to be formed by using any two of these cards. Two cards are picked at random, one after another, without replacement.

**(a)**List all sample space

**(b)**List all the outcomes of the events and find the probability that

**(i)**The code begins with the letter

*P*.

**(ii)**The code consists of two vowel or two consonants.

*
*

Solution:Solution:

**(a)**

Sample space,

*S*= {(

*G*,*R*), (*G*,*A*), (*G*,*P*), (*G*,*E*), (*R*,*G*), (*R*,*A*), (*R*,*P*), (*R*,*E*), (*A*,*G*), (*A*,*R*), (

*A*,*P*), (*A*,*E*), (*P*,*G*), (*P*,*R*), (*P*,*A*), (*P*,*E*), (*E*,*G*), (*E*,*R*), (*E*,*A*), (*E*,*P*)}

(b)

(b)

*n*(

*S*) = 20

Let

*A*= Event of choosing a code begins with the letter

*P*

*B*= Event of choosing the code consists of two vowel or two consonants.

(i)

(i)

*A*= {(

*P*,

*G*), (

*P*,

*R*), (

*P*,

*A*), (

*P*,

*E*)}

*n*(

*A*) = 4

$P\left(A\right)=\frac{4}{20}=\frac{1}{5}$

(ii)

(ii)

*B*= {(

*G*,

*R*), (

*G*,

*P*), (

*R*,

*G*), (

*R*,

*P*), (

*A*,

*E*), (

*P*,

*G*), (

*P*,

*R*), (

*E*,

*A*)}

*n*(

*B*) = 8