Find the probability that the new course is introduced.

Question: Three professors A, B and C appear in an interview for the post of Principal. Their chances of getting selected as a principal are 2/9, 4/9, 1/3. The probabilities they introduce new course in the college are 3/10, 1/2, 4/5 respectively. Find the probability that the new course is introduced.

(you are supposed to fill in the blanks for this question.)

Let A, B, C be the events that prof. A, B and C are selected as principal.

Given P(A) = 2/9 , P(B) = 4/9 , P(C) = 1/3 = 3/9

Let N be the event that New Course is introduced

P(N|A) = ___, P(N|B) = ___, P(N|C) = 4/5

N = (A ∩ N) U (B ∩ __) U (__ ∩ N)

P(N) = (A ∩ N) + (B ∩ __) + (__ ∩ N)

= P(A) x P(N|A) + P(__) x ___ + ___ x P(N|C)

= ___ ___ + ___ ___ + ___ ___

= ___ + ___ + ___

= ___    

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Background knowledge/info: 

1. Multiplication theorem: The probability of simultaneous occurrence of two events A and B is equal to the product of the probability of the other, given that the first one has occurred. 

P(A ∩ B)  = P(A) . P(B|A)
P(B|A) = P(A ∩ B) / P(A)

Similarly,
P(A ∩ B)  = P(B) . P(A|B)
P(A|B) = P(A ∩ B) / P(B)

2. Independent events: The events that do NOT depend on (or affect the occurrence of) each other. The probability of one event is NOT changed by the realization or outcome of another event. 

If A and B are independent events, then 
P(A|B) = P(A|B') = P(A) 
P(B|A) = P(B|A') = P(B)

P(A ∩ B)  = P(A) x P(B)