Are Jack and Jill making their weather predictions independently?...

[pru_reardon]

...Mathematically justify your answer.? QUESTION:
Jack and Jill are two weather forcasters in Gonzales. The probability that Jack accurately predicts the weather on any given day is 0.57, and the probability that Jill accurately predicts the weather on any given day is 0.62. If the probability at AT LEAST ONE of them is correct on any given day is 0.81, are Jack and Jill making their weather predictions independently? Mathematically justify your answer.

MY ANSWER:
J-Jack
D-Jill

I know that the test for independent events is P(J intersection D) = P(J) * P(D)
We were given P(J union D) which was 0.81... correct?
So I took the complement of each one P(J) and P(D) and got.. P(Jc)=0.43 and P(Dc)=0.38.
Then I said that P(Jc) * P(Dc) = 0.1634 and 1-.1634=0.8366 NOT 0.81. The answer I got for (J intersection D) was 0.38 and therefore:

P(J intersection D)=P(J) * P(D) or
0.38 DOES NOT = (.0.57) * (0.62) which makes their weather predictions dependent and not independent.

Did I do this correctly? If not, where did I go wrong? Thanks in advance for your help!!