Credit to Sanjitscorps18 for this answer. He sent it to my PM. But, this is so nicely written that it should be here and in PM.
S consists of {A, B, any other outcomes}
Now if we visualize this from a Venn diagram perspective, there is a possibility that A and B can have an overlap. Now using this in eq(1) we get
S (Total outcomes) = Outcomes (A) + Outcomes (B) + Outcomes (A and B) + Outcomes (Others)
S (Total outcomes) = Outcomes (A) + Outcomes (B) + Outcomes (A and B) + Outcomes (Neither A nor B)
When we take probability of the above equation we get
S (Total Probability) = Probability(A) + Probability(B) + Probability(A and B) + Probability(Neither A nor B)
=> 1 = P(A) + P(B) + P(A and B) + P(neither) ----- (1)
Essentially we need the value of P(A) + P(A and B) as the question asks what is the probability of A
Now taking the individual statements
1. P(AUB)=0.7
P(A) + P(B) + P(A and B) = 0.7
=> P(neither) = 0.3
(2) P(AU~B)=0.9
P(A) + P(A and B) + P(neither) = 0.9
=> P(B) = 0.1
Clearly, we cannot find P(A) + P(A and B) from either of the statements alone
Substituting the values in equation (1) above we get
1 = P(A) + P(B) + P(A and B) + P(neither)
=> 1 = P(A) + 0.1 + P(A and B) + 0.3
=> P(A) + P(A and B) = 1 - 0.3 - 0.1 = 0.6
=> P(A) + P(A and B) = 0.6
_________________
~In Scientia Opportunitas~