What is P(AUBUC) formula? Which of the following are possible values of  P(AUBUC) ?

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What is P(AUBUC) formula?

Question : Suppose are events in a common probability space with

P(A) =0.2 , P(B) =0.2, P(C) =0.3, P(A∩B) =0.1 , P(A∩C) =0.1, P(B∩C) = 0.1

Which of the following are possible values of  P(AUBUC) ?

1. 0.5

2. 0.3

3. 0.4

4. 0.9

Solution :

Given values of the probabilities

P(A) = 0.2, P(B) = 0.2,  P(C) =0.3, P(A ∩ B) = 0.1,  P(A ∩ C)  = 0.1,  P(B ∩ C) = 0.1

To find the value of P(AUBUC)

We shall use the P(AUBUC) formula

P(AUBUC) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)

Substituting values

P(AUBUC) = 0.2 + 0.2 + 0.3 – 0.1 – 0.1 – 0.1 + P(A ∩ B ∩ C)

P(AUBUC) = 0.4 + P(A ∩ B ∩ C)

Since probability of any event is lying between 0 to 1. That is

0≤P(A ∩ B ∩ C)≤1

0.4≤0.4+P(A ∩ B ∩ C)≤0.6

0.4≤P(AUBUC)≤0.6

Therefore possible values of P(AUBUC) are 0.4, 0.5

Therefore options (1) and (3) are correct.


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