QUESTIONS ON ADDITION THEOREM OF PROBABILITY

Question 1 :

If P (A) = 2/3, P(B) = 2/5, P(AU B) = 1/3 then find P(AnB).

Solution :

P(A n B)  =  P(A) + P(B) - P(AUB)

P(A n B)  =  (2/3) + (2/5) - (1/3)

P(A n B)  =  (10 + 6 - 5)/15

P(A n B)  =  11/15

Question 2 :

A and B are two events such that, P(A) = 0.42, P(B) = 0.48 and P(A n B) = 0.16. Find

(i) P (not A) (ii) P (not B) (iii) P (A or B)

Solution :

P(A) = 0.42

P(not A)  =  1 - P(A) 

  =  1 - 0.42

P(A bar)  =  0.58

(ii) P (not B)

P(not B)  =  1 - P(B) 

  =  1 - 0.48

P(not B)  =  0.52

(iii) P (A or B)  =  P(A U B)

  =  P(A) + P(B) - P(A n B)

  =  0.42 + 0.48 - 0.16

  =  0.74

Question 3 :

If A and B are two mutually exclusive events of a random experiment and P(not A) = 0.45, P(A U B) = 0.65, then find P(B)

Solution :

Since A and B are mutually exclusive events, P(A n B)  =  0.

P(not A) = 0.45

P(A)  =  1 - P(not A) 

P(A)  =  1 - 0.45

P(A)  =  0.55

P(An B) =  P(A) + P(B) - P(AUB)

0  =  0.55 + P(B) - 0.65

0.1  =  P(B)

P(B)  =  0.1

Question 4 :

The probability that atleast one of A and B occur is 0.6. If A and B occur simultaneously with probability 0.2, then find P(A bar) + P(B bar).

Solution : 

P(A or B)  =  P(A U B)  =  0.6

P(A n B)  =  0.2

P(A n B)  =  P(A) + P(B) - P(A U B)

0.2  =  P(A) + P(B) - 0.6

P(A) + P(B)  =  0.2 + 0.6

P(A) + P(B)  =  0.8

1 - P(not A) + 1 - P(not B)  =  0.8

2 - 0.8  =  P(not A) + P(not B)

P(not A) + P(not B)  =  1.2

Question 5 :

The probability of happening of an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then find the probability that neither A nor B happen

Solution :

P(A)  =  0.5, P(B)  =  0.3

P(A n B) = 0(Since A and B are mutually exclusive events)

P(A bar n B bar) 

P(A U B)  =  P(A) + P(B) - P(A n B)

P(A U B)  =  0.5 + 0.3 - 0

P(A U B)  =  0.8

P[(AUB) whole bar]  = 1 - P(A U B)

P[(AUB) whole bar]  = 1 - 0.8

P[(AUB) whole bar]  = 0.2

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