Given below are two statement based on the following
If A and B are independent events such that P(A)=p, P(B)=2p and P (exactly one of A,B) = $$\dfrac{5}{9}$$
Statement I: p = $$\frac{1}{3}$$
Statement II: p = $$\frac{5}{12}$$
In the light of the above statements, choose the correct answer form the question given below

$$P(A)=p$$
$$P(B)=2p$$

P(Exactly one of A or B) = $$\dfrac{5}{9}$$. This means that either A happens and B does not happen, or B happens and A does not happen. Therefore - 

$$P\left(\overline{A}\right)P\left(B\right)+P\left(\overline{B}\right)P\left(A\right)=\dfrac{5}{9}$$

$$(1-p)(2p)+(1-2p)(p)=\dfrac{5}{9}$$

$$3p-4p^2=\dfrac{5}{9}$$

$$36p^2-27p+5=0$$

$$36p^2-12p-15p+5=0$$

$$12p\left(3p-1\right)-5\left(3p-1\right)=0$$

$$\left(12p-5\right)\left(3p-1\right)=0$$

$$p=\dfrac{5}{12}\ or\ p=\dfrac{1}{3}$$

Thus, both statements are true.