WORD PROBLEMS BASED ON ADDITION THEOREM OF PROBABILITY

Problem 1 :

Two dice are rolled once. Find the probability of getting an even number on the first die or a total of face sum 8.

Solution :

S = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Let "A" be the event of getting an even number on the first die

A = {(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

n(A)  =  18

P(A)  =  n(A)/n(S)

P(A)  =  18/36

Let "B" be the event of a total of face sum 8

B = {(2, 6) (3, 5) (4, 4) (5, 3) (6, 2)}

n(B)  =  5

P(B)  =  n(B)/n(S)  

P(B)  =  5/36

A n B  = {(2, 6) (4, 4) (6, 2)}

n(A n B)  =  3

P(A n B)  =  n(AnB)/n(S)

P(A n B)  =  3/36

P(A U B)  =  P(A) + P(B) - P(AnB)

P(A U B)  =  (18/36) + (5/36) - (3/36)

P(A U B)  =  (18 + 5 - 3)/36

P(A U B)  =  20/36   =  5/9

Problem 2 :

From a well-shuffled pack of 52 cards, a card is drawn at random. Find the probability of it being either a red king or a black queen.

Solution :

Total number of cards n(S)  =  52

Let "A" be the event of getting red king card.

n(A)  =  2

P(A)  =  n(A) / n(S)  =  2/52

Let "B" be the event of getting black queen card.

n(B)  =  2

P(B)  =  n(B) / n(S)  =  2/52

Since A and B are mutually exclusive events, A n B  =  0

P(A U B)  =  P(A) + P(B) - P(A n B) 

P(A U B)  =  (2/52) + (2/52) - 0

P(A U B)  =  4/52 

P(AUB)  =  1/13

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