One shot or at least two shorts in three games?

You have a basketball hoop and someone says that you can play 1 of 2 games.
Game #1: You get one shot to make the hoop.
Game #2: You get three shots and you have to make 2 of 3 shots.
If p is the probability of making a particular shot, for which values of p should you pick one game or the other?

My initial thoughts (Solution):
For game #1, you have probability lp of winning. For game #2, you can either make 2 shots of 3, with probability 3p^2(1-p), or make all of the three shots, with probability p^3. Therefore, to choose game #1, you need to have: p<3p^2(1-p)+p^3, which gives us p<1/2.


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