Ones and Twos, Twos and Threes

Organized basketball has some different scoring rules from pickup basketball. In organized basketball, a regular shot is worth two points and a long shot is worth three. In pickup, a regular shot is worth one and a long shot is worth two. Also, in pickup half-court games, most people play that the team that made a shot gets the ball again ("winners"), while in full-court games (pickup and organized), the team that made a shot does not get the ball ("losers").

This has some strategic implications. To take an extreme case, suppose you're playing with 1's and 2's, and ignore offensive rebounding and turnovers. In losers, a 51% chance at a 2-pointer is better than a 100% chance at a 1-pointer. In winners though, if you can make a 1-pointer 100% of the time and you can get the ball, you always win.

Suppose you can make a regular shot 50% of the time, and again, assume no offensive rebounds or turnovers. What percentage do you need to shoot a long shot instead?

If you're playing with 1's and 2's, losers, it's 25%
With 1's and 2's, winners, it's 33%
With 2's and 3's, losers, its 33%

If you can make a regular shot 33% of the time:

If you're playing with 1's and 2's, losers, it's 16.7%
With 1's and 2's, winners, it's 20%
With 2's and 3's, losers, its 22%

As a side note, if you're in an NBA game, you should generally feel indifferent between a 3-pointer and a 2-pointer if you're 50% more likely to make the 2. But if you deciding between a 2 and a 3 at the very end of a game, making a 3 means that you definitely win, where making a 2 means that you're going to overtime. So assuming that you think you have a 50-50 shot in overtime, you need to be twice as likely to make the 2 as the 3.