INDEPENDENT EVENTS OR NOT MUTUALLY EXCLUSIVE EVENTS

 PROBLEMS BASED ON INDEPENDENT EVENTS OR NOT MUTUALLY EXCLUSIVE EVENTS

$P(A \cup B)=P(A)+P(B)-P(A \cap B)$ (OR) $ P(A \cap B)=P(A).P(B)$

EXAMPLE 1
 One  card drawn from a deck of 52 cards. what is the probability of the  card being either a red or a king 

Solution:  Given total number of card is 52

$  n(S)=52 $

Let A be a event that the card drawn is red
$ n(A)=26 $
$ P(A)=\frac{n(A)}{n(S)}=\frac{26}{52}=\frac{1}{2} $

Let B be a event that the card drawn is king

$ n(B)=4 $
$P(B)=\frac{n(B)}{n(S)}=\frac{4}{52}=\frac{1}{13} $

Here $A \cap   B$  there are red colored of king card

$ n(A\cap B)=2 $
$ P(A \cap B )= \frac{n(A\cap B)}{n(S)} =\frac{2}{52}=\frac{1}{26} $
.
$ P(A \cup  B) = P(A)+P(B)- P(A \cap B ) $          [   A  and B  are  not mutually exclusive  events]
                    $ = \frac{1}{2}+\frac{1}{13}-\frac{1}{26} $                             [$ P(A)=\frac{1}{13}$ P(B)=$ \frac{1}{13}$ ]
                     $ =  \frac{13+2-1}{26}$
                        $ = \frac{14}{26}$
$ P(A \cup  B) = \frac{7}{13} $
 

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EXAMPLE 2:
A is known to hit the target in 2 out of 5 shots whereas B is known to hit the target in 3 out of  4 shots.Find the probability of the target being hit when they both try?

Solution:
Let A be a event of that 'A' hit the target

$ P(A)=\frac{2}{5}  $                     Given

Let B be a event of that 'B' hit the target

$  P(B)=\frac{3}{4} $                      Given

$ P(A \cup B) = P(A)+P(B)-P(A \cap B) $
                    $ =  P(A)+P(B)-P(A) .P( B)  $                 [  A  and  B   are independent   event]
                    $ = \frac{2}{5}+\frac{3}{4}-\left( \frac{2}{5}.\frac{3}{4} \right) $
                    $ = \frac{8+15}{20}-\frac{6}{20}$
                    $ = \frac{23-6}{20} $
    $ P(A \cup B) = \frac{17}{20} $

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