I believe the answer is yes, with two possible explanations.
The first explanation deals with a straightforward expected value calculation. Because we have uncertainty about the lottery, we can calculate the expected utility for both events.
The expected utility of the P300 is certain:
U(300) = 300^2
For the lottery, the expected utility can be calculated with the probability of winning (because in the case of losing utility is zero:
U = .2 * U(1,000)
= .2 * 1,000,000
So, expected utility of the lottery is higher, and Adele would buy the lottery ticket.
The second explanation is that Adele's utility function is the square of the amount of money she receives. This is exponentially increasing, which means her marginal utility of money is also an increasing function. (I am rusty with calculus, but I believe her marginal utility is 2x, where x is the input to the utility function.) Because she has increasing marginal utility of money, this also makes her risk seeking, and she will buy the lottery ticket.
I'm not 100% confident in this response, but I'm pretty confident that I have most of the correct principles.