2-pointer and-one obviously gets a foul on an opposing player but you are not guaranteed to make the free throw, while a 3-pointer is guaranteed 3 points.

Which would be more advantageous to a team, specifically in a NBA game where the free throw conversion rate is around 75%?

It seems like a bucket and-one is better just by observing player celebration.

  • Hi Bakalolo, welcome to the site. What do you mean by "more advantageous" specifically? Both are 3 points once they're fully scored, other team gets the ball, so there's no difference in the outcome other than the foul. Are you asking to compare a "made 3 pointer" to a "made 2 pointer with a foul", not assuming the made and-1 yet? Or are you really asking, "why do they celebrate more after a 2 pointer with a foul, than after a 3 pointer"? Or are you asking "is the 75% chance of a 3rd point + a player foul worth more than 100% of a 3rd point + no foul"? – Joe Aug 31 '20 at 15:20
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    The player celebration part may be due to a couple of reasons * A foul on the opposing team is an advantage * What was a 2 point attempt can now be converted into a 3. – Rahul Kadukar Aug 31 '20 at 18:19
  • @Joe I am asking which scenario would give the team a better chance to win in general – bakalolo Sep 1 '20 at 16:55

There's not a perfect answer to this, particularly given NBA advanced stats are a little rougher than some other sports (MLB, NFL), but we are comparing these two possibilities:

Three pointer made: 3 points

2 pointer made, and-one opportunity: 2 points, and one FTA, and one personal foul

Fouls in the NBA's PER (Player Efficiency Rating) are rated as such:

-PF * ((lg_FT / lg_PF) - 0.44 * (lg_FTA / lg_PF) * VOP)

VOP is "value of possession", roughly how much a team is estimated to score on a possession - around a point, give or take, depending on the year. This means they consider a foul to be worth negative 1 points times (free throws made / total personal fouls), which is the made-free-throw rate, and then add back in basically the percentage of free throws that end the possession for the other team.

This is pretty basic, and not unfortunately very useful for this - because we have more information than this formula does. But it gives us two things:

  1. Personal fouls aren't considered to have any specific value in NBA advanced statistics - meaning, there's no "penalty" for committing a foul other than the points you on average give up. You can argue there probably should be one, but given they don't have a penalty for it in PER, it's not something we can reasonably determine.
  2. The value for the "and one" should be (2) + (FT%) + (1-FT%)(OREB%)(VOP)

Given OP asked us to assume FT% is 0.75 (close enough), and OREB% is around 22% for the league this year, and VOP is around 1.06 this year for the average team, we can calculate the value of an and-one to be:

2 + 0.75 + (.25*1.06*.22) = 2.81

So an 'and-one' is worth a bit less than a made 3 pointer, analytically.

Of course, there's no question that an and-one that gets a sixth foul on Anthony Davis is worth a lot more than the 2.81 points here; but as far as I can tell, there's no rigorous method for determining that value.


Taking out of the equation, the likeliness that either field goal will be scored, the 3-point shot is more advantageous.

In sports, you always prefer to keep what is given. The 3-point shot leaves no points up to chance. With the and-1, you are further gambling to gain the extra point. An average of 75% odds that the extra point will be scored is not a desirable outcome compared to 100% with the 3-point shot.

There are other considerations with this analysis though. In general, the 3-point shot is a lower-probability shot compared to most 2-point shots. And as mentioned above, it may instead be desirable to draw the foul, depending on team-foul situation and whether the defender has a high number of fouls already.

Putting these considerations aside, it is always preferred to get points without leaving it up to chance.

  • The entire purpose of the question is to establish whether those other considerations outweigh the surety of the additional point. Putting them aside entirely means the question is not being answered. – Nij Sep 1 '20 at 1:55
  • I put them aside because they are negligible; points on the board supersede any other considerations. – Jason P Sallinger Sep 1 '20 at 13:58
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    That's not what the question asked though. You need to demonstrate that no other possible combination of other factors affecting the rest of the game could be worth more than one extra point. I can see one obvious situation where they would, when the foul results in ejection of a good player, points expectation over the next M minutes can very definitely be greater than a single point now. So this answer at best is wrong. – Nij Sep 2 '20 at 1:48

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