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.