There are two Half Dome permit lotteries, not one. The preseason lottery takes applications on Recreation.gov through March with results in mid-April; the daily lottery runs every day the cables are up, taken two days before the hike date. In the most recent published season about one preseason application in five succeeded, weekdays drew meaningfully better odds than weekends, and late-season weekdays were the best draw of the year. Strategy: use all seven date choices, front-load unpopular dates, enter both lotteries, and plan the hike mid-trip so the daily lottery gets several independent draws.
The full mechanics: the complete lottery guide. Whether the hike itself is the right goal: So You Want to Hike Half Dome.