Excellent question. I've only got some ideas at the minute rather than a firm answer.
I think there are two parts to this. Firstly is the part of getting the tickets chosen numbers on the blockchain.
This will be the easier part , as when the ticket is purchased the numbers they've chosen can be coded into the transaction on the blockchain.
The second part is getting the Powerball numbers onto the blockchain.
One possible idea is using a consensus model similar to what Augur (and I think Gnosis) are planning on doing. Users confirm winning numbers. However to me this is open to fraud, especially when there are low numbers of ticket sales.
An easier (and better?) option would be that we pull the winning numbers from the Powerball website and create a transaction that is entered onto the blockchain. This could be combined with manual human verification.
We could also pull the powerball winning numbers from the state lottery websites that sell Powerball tickets e.g. http://www.txlottery.org/export/sites/lottery/Games/Powerball/Winning_Numbers/details.html_1130671244.html and http://louisianalottery.com/powerball/tab/winning-numbers/2017-06-21 etc.
This would make it more secure, as somone who hacked the Powerball website (unlikely but possible) would also need to hack all the state run lottery websites if they wanted to input false numbers.
Thoughts?