Method

The Process

1 Get Every 3-Point Attempt

We fetch play-by-play data and extract each 3-point shot with its context:

Shot type: Catch-and-shoot vs Pull-up (stepback, running, etc.)
Player history: Career 3PA/3PM with recency weighting
Free throw percentage: Used to anchor the prior expectation

This game had 70 total 3-point attempts.

2 Calculate Each Player's Expected 3P% Using FT%-Based Prior

We use a Bayesian Beta-Binomial model where the prior is anchored to the player's free throw percentage. FT% is a stable indicator of shooting skill that predicts 3PT ability.

Step 1: Calculate FT%-based prior mean Prior 3P% = intercept + slope x FT% Context-specific parameters (calibrated from 10 seasons of data): Overall: prior = 0.207 + 0.193 x FT% kappa = 437 Catch-and-shoot: prior = 0.199 + 0.221 x FT% kappa = 418 Pull-up: prior = 0.174 + 0.207 x FT% kappa = 270 Step 2: Combine prior with observed shooting (Beta-Binomial) alpha = (prior x kappa) + weighted_makes beta = ((1 - prior) x kappa) + (weighted_attempts - weighted_makes) Step 3: Posterior expected 3P% Expected 3P% = alpha / (alpha + beta)

Worked Examples: Veteran vs Rookie

Player	FT%	Prior	kappa	Weighted 3P	alpha	beta	Expected 3P%
T. Maxey (veteran)	87.9%	37.7%	437	911.9/2405.3	1076.6	1766.0	37.9%
V. Edgecombe (rookie)	75.0%	35.2%	437	203.4/562.5	357.2	642.6	35.7%

T. Maxey: (87.9% FT shooter, high-volume veteran) Prior = 0.207 + 0.193 x 0.879 = 0.377 (37.7%) alpha = (0.377 x 437) + 911.9 = 164.7 + 911.9 = 1076.6 beta = (0.623 x 437) + (2405.3 - 911.9) = 272.3 + 1493.4 = 1766.0 Expected = 1076.6 / (1076.6 + 1766.0) = 37.9% V. Edgecombe: (75% FT shooter, rookie with limited data) Prior = 0.207 + 0.193 x 0.750 = 0.352 (35.2%) alpha = (0.352 x 437) + 203.4 = 153.8 + 203.4 = 357.2 beta = (0.648 x 437) + (562.5 - 203.4) = 283.2 + 359.1 = 642.6 Expected = 357.2 / (357.2 + 642.6) = 35.7% Note: The FT%-based prior gives Maxey a higher baseline (37.7%) than Edgecombe (35.2%) because better free throw shooters tend to be better 3PT shooters.

3 Apply Context-Specific Adjustments

For each shot, we adjust the player's expected % based on shot type:

Catch-and-shoot shots: Higher expected % (easier shots) C&S prior = 0.199 + 0.221 x FT% Pull-up shots (stepback, running, etc.): Lower expected % (harder shots) Pull-up prior = 0.174 + 0.207 x FT% Example for T. Maxey (87.9% FT): C&S expected: ~39.5% (prior = 39.3%) Pull-up expected: ~35.8% (prior = 35.6%)

4 Sum Expected Makes & Calculate Adjustment

Sum up expected make probabilities for each team, compare to actual makes:

Luck = Actual Makes - Expected Makes Point Adjustment = 3 x Luck - (ORB Rate x PPP x |Luck|) = 2.7 points per make above/below expectation

MIN Shot-by-Shot Breakdown

Player	Result	Shot Type	FT%	Weighted 3P	Posterior	Final Exp%
J. Randle	X Miss	Catch Shoot	80.6%	937.7/2826.2	33.6%	34.9%
A. Edwards	+ Made	Catch Shoot	83.7%	1256.8/3376.4	37.2%	38.7%
A. Edwards	X Miss	Pullup	83.7%	1256.8/3376.4	37.2%	35.0%
T. Shannon Jr.	X Miss	Catch Shoot	81.0%	40.6/105.9	36.7%	38.2%
D. DiVincenzo	+ Made	Catch Shoot	77.8%	1161.0/3060.7	37.7%	39.2%
J. McDaniels	X Miss	Pullup	81.3%	576.5/1558.9	36.9%	34.7%
A. Dosunmu	+ Made	Catch Shoot	78.5%	555.8/1401.4	38.8%	40.2%
A. Edwards	X Miss	Stepback	83.7%	1256.8/3376.4	37.2%	35.0%
J. McDaniels	X Miss	Catch Shoot	81.3%	576.5/1558.9	36.9%	38.4%
B. Hyland	X Miss	Stepback	88.5%	570.5/1538.9	37.2%	35.1%
B. Hyland	X Miss	Catch Shoot	88.5%	570.5/1538.9	37.2%	38.9%
J. Randle	+ Made	Catch Shoot	80.6%	937.7/2826.2	33.6%	34.9%
D. DiVincenzo	X Miss	Stepback	77.8%	1161.0/3060.7	37.7%	35.4%
J. Randle	X Miss	Pullup	80.6%	937.7/2826.2	33.6%	31.6%
J. Clark	X Miss	Running	78.4%	62.7/192.1	34.9%	32.8%
A. Edwards	+ Made	Stepback	83.7%	1256.8/3376.4	37.2%	35.0%
A. Edwards	+ Made	Running	83.7%	1256.8/3376.4	37.2%	35.0%
A. Edwards	X Miss	Catch Shoot	83.7%	1256.8/3376.4	37.2%	38.7%
M. Conley	X Miss	Catch Shoot	90.0%	2053.0/5335.5	38.4%	40.1%
D. DiVincenzo	+ Made	Catch Shoot	77.8%	1161.0/3060.7	37.7%	39.2%
D. DiVincenzo	+ Made	Catch Shoot	77.8%	1161.0/3060.7	37.7%	39.2%
A. Dosunmu	X Miss	Catch Shoot	78.5%	555.8/1401.4	38.8%	40.2%
J. Randle	X Miss	Catch Shoot	80.6%	937.7/2826.2	33.6%	34.9%
A. Edwards	X Miss	Stepback	83.7%	1256.8/3376.4	37.2%	35.0%
D. DiVincenzo	X Miss	Catch Shoot	77.8%	1161.0/3060.7	37.7%	39.2%
J. McDaniels	X Miss	Catch Shoot	81.3%	576.5/1558.9	36.9%	38.4%
J. McDaniels	X Miss	Stepback	81.3%	576.5/1558.9	36.9%	34.7%
A. Edwards	X Miss	Pullup	83.7%	1256.8/3376.4	37.2%	35.0%
B. Hyland	+ Made	Running	88.5%	570.5/1538.9	37.2%	35.1%
M. Conley	X Miss	Catch Shoot	90.0%	2053.0/5335.5	38.4%	40.1%
J. Randle	X Miss	Catch Shoot	80.6%	937.7/2826.2	33.6%	34.9%
B. Hyland	X Miss	Catch Shoot	88.5%	570.5/1538.9	37.2%	38.9%
A. Dosunmu	+ Made	Running	78.5%	555.8/1401.4	38.8%	36.5%
TOTALS		12.3 exp
Actual: 10 made		-2.3 luck

Player

Result

Shot Type

FT%

Weighted 3P

Posterior

Final Exp%

J. Randle

X Miss

Catch Shoot

80.6%

937.7/2826.2

33.6%

34.9%

A. Edwards

+ Made

Catch Shoot

83.7%

1256.8/3376.4

37.2%

38.7%

A. Edwards

X Miss

Pullup

83.7%

1256.8/3376.4

37.2%

35.0%

T. Shannon Jr.

X Miss

Catch Shoot

81.0%

40.6/105.9

36.7%

38.2%

D. DiVincenzo

+ Made

Catch Shoot

77.8%

1161.0/3060.7

37.7%

39.2%

J. McDaniels

X Miss

Pullup

81.3%

576.5/1558.9

36.9%

34.7%

A. Dosunmu

+ Made

Catch Shoot

78.5%

555.8/1401.4

38.8%

40.2%

A. Edwards

X Miss

Stepback

83.7%

1256.8/3376.4

37.2%

35.0%

J. McDaniels

X Miss

Catch Shoot

81.3%

576.5/1558.9

36.9%

38.4%

B. Hyland

X Miss

Stepback

88.5%

570.5/1538.9

37.2%

35.1%

B. Hyland

X Miss

Catch Shoot

88.5%

570.5/1538.9

37.2%

38.9%

J. Randle

+ Made

Catch Shoot

80.6%

937.7/2826.2

33.6%

34.9%

D. DiVincenzo

X Miss

Stepback

77.8%

1161.0/3060.7

37.7%

35.4%

J. Randle

X Miss

Pullup

80.6%

937.7/2826.2

33.6%

31.6%

J. Clark

X Miss

Running

78.4%

62.7/192.1

34.9%

32.8%

A. Edwards

+ Made

Stepback

83.7%

1256.8/3376.4

37.2%

35.0%

A. Edwards

+ Made

Running

83.7%

1256.8/3376.4

37.2%

35.0%

A. Edwards

X Miss

Catch Shoot

83.7%

1256.8/3376.4

37.2%

38.7%

M. Conley

X Miss

Catch Shoot

90.0%

2053.0/5335.5

38.4%

40.1%

D. DiVincenzo

+ Made

Catch Shoot

77.8%

1161.0/3060.7

37.7%

39.2%

D. DiVincenzo

+ Made

Catch Shoot

77.8%

1161.0/3060.7

37.7%

39.2%

A. Dosunmu

X Miss

Catch Shoot

78.5%

555.8/1401.4

38.8%

40.2%

J. Randle

X Miss

Catch Shoot

80.6%

937.7/2826.2

33.6%

34.9%

A. Edwards

X Miss

Stepback

83.7%

1256.8/3376.4

37.2%

35.0%

D. DiVincenzo

X Miss

Catch Shoot

77.8%

1161.0/3060.7

37.7%

39.2%

J. McDaniels

X Miss

Catch Shoot

81.3%

576.5/1558.9

36.9%

38.4%

J. McDaniels

X Miss

Stepback

81.3%

576.5/1558.9

36.9%

34.7%

A. Edwards

X Miss

Pullup

83.7%

1256.8/3376.4

37.2%

35.0%

B. Hyland

+ Made

Running

88.5%

570.5/1538.9

37.2%

35.1%

M. Conley

X Miss

Catch Shoot

90.0%

2053.0/5335.5

38.4%

40.1%

J. Randle

X Miss

Catch Shoot

80.6%

937.7/2826.2

33.6%

34.9%

B. Hyland

X Miss

Catch Shoot

88.5%

570.5/1538.9

37.2%

38.9%

A. Dosunmu

+ Made

Running

78.5%

555.8/1401.4

38.8%

36.5%

TOTALS

12.3 exp

Actual: 10 made

-2.3 luck

MIN Player Summary

Player	FT%	Shots	Made	Expected Makes	Luck	Point Impact
D. DiVincenzo	77.8%	5	3	1.93	+1.07	+2.9
A. Dosunmu	78.5%	3	2	1.17	+0.83	+2.2
A. Edwards	83.7%	8	3	2.87	+0.13	+0.4
J. Clark	78.4%	1	0	0.33	-0.33	-0.9
T. Shannon Jr.	81.0%	1	0	0.38	-0.38	-1.0
B. Hyland	88.5%	4	1	1.48	-0.48	-1.3
J. Randle	80.6%	5	1	1.71	-0.71	-1.9
M. Conley	90.0%	2	0	0.80	-0.80	-2.2
J. McDaniels	81.3%	4	0	1.46	-1.46	-3.9
TOTAL		33	10	12.14	-2.14	-5.8

Player

FT%

Shots

Made

Expected Makes

Luck

Point Impact

D. DiVincenzo

77.8%

1.93

+1.07

+2.9

A. Dosunmu

78.5%

1.17

+0.83

+2.2

A. Edwards

83.7%

2.87

+0.13

+0.4

J. Clark

78.4%

0.33

-0.33

-0.9

T. Shannon Jr.

81.0%

0.38

-0.38

-1.0

B. Hyland

88.5%

1.48

-0.48

-1.3

J. Randle

80.6%

1.71

-0.71

-1.9

M. Conley

90.0%

0.80

-0.80

-2.2

J. McDaniels

81.3%

1.46

-1.46

-3.9

TOTAL

12.14

-2.14

-5.8

PHI Shot-by-Shot Breakdown

Player	Result	Shot Type	FT%	Weighted 3P	Posterior	Final Exp%
K. Oubre Jr.	+ Made	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
T. Maxey	+ Made	Catch Shoot	87.9%	911.9/2405.3	37.9%	39.5%
D. Barlow	X Miss	Running	63.6%	45.2/166.0	31.4%	29.1%
V. Edgecombe	+ Made	Catch Shoot	75.0%	203.4/562.5	35.7%	37.0%
T. Maxey	X Miss	Pullup	87.9%	911.9/2405.3	37.9%	35.8%
V. Edgecombe	X Miss	Stepback	75.0%	203.4/562.5	35.7%	33.3%
Q. Grimes	+ Made	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
V. Edgecombe	+ Made	Catch Shoot	75.0%	203.4/562.5	35.7%	37.0%
Q. Grimes	+ Made	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
Q. Grimes	X Miss	Stepback	75.7%	829.0/2255.1	36.5%	34.2%
T. Maxey	X Miss	Stepback	87.9%	911.9/2405.3	37.9%	35.8%
T. Maxey	+ Made	Pullup	87.9%	911.9/2405.3	37.9%	35.8%
Q. Grimes	X Miss	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
C. Payne	+ Made	Pullup	90.7%	704.1/1939.0	36.7%	34.7%
K. Oubre Jr.	X Miss	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
K. Oubre Jr.	+ Made	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
C. Payne	X Miss	Pullup	90.7%	704.1/1939.0	36.7%	34.7%
T. Maxey	X Miss	Stepback	87.9%	911.9/2405.3	37.9%	35.8%
V. Edgecombe	+ Made	Catch Shoot	75.0%	203.4/562.5	35.7%	37.0%
V. Edgecombe	+ Made	Catch Shoot	75.0%	203.4/562.5	35.7%	37.0%
K. Oubre Jr.	X Miss	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
K. Oubre Jr.	X Miss	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
Q. Grimes	+ Made	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
V. Edgecombe	+ Made	Pullup	75.0%	203.4/562.5	35.7%	33.3%
J. Walker	X Miss	Catch Shoot	69.0%	122.3/396.3	32.5%	33.6%
Q. Grimes	X Miss	Stepback	75.7%	829.0/2255.1	36.5%	34.2%
C. Payne	X Miss	Pullup	90.7%	704.1/1939.0	36.7%	34.7%
T. Maxey	+ Made	Pullup	87.9%	911.9/2405.3	37.9%	35.8%
V. Edgecombe	+ Made	Stepback	75.0%	203.4/562.5	35.7%	33.3%
Q. Grimes	+ Made	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
K. Oubre Jr.	X Miss	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
K. Oubre Jr.	+ Made	Catch Shoot	75.1%	1149.0/3502.0	33.1%	34.4%
T. Maxey	+ Made	Pullup	87.9%	911.9/2405.3	37.9%	35.8%
Q. Grimes	+ Made	Catch Shoot	75.7%	829.0/2255.1	36.5%	37.9%
J. Edwards	+ Made	Catch Shoot	69.6%	140.2/400.4	34.6%	35.8%
J. Edwards	X Miss	Running	69.6%	140.2/400.4	34.6%	32.3%
D. Terry	+ Made	Pullup	71.0%	119.2/343.7	34.5%	32.2%
TOTALS		12.9 exp
Actual: 21 made		+8.1 luck

Player

Result

Shot Type

FT%

Weighted 3P

Posterior

Final Exp%

K. Oubre Jr.

+ Made

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

T. Maxey

+ Made

Catch Shoot

87.9%

911.9/2405.3

37.9%

39.5%

D. Barlow

X Miss

Running

63.6%

45.2/166.0

31.4%

29.1%

V. Edgecombe

+ Made

Catch Shoot

75.0%

203.4/562.5

35.7%

37.0%

T. Maxey

X Miss

Pullup

87.9%

911.9/2405.3

37.9%

35.8%

V. Edgecombe

X Miss

Stepback

75.0%

203.4/562.5

35.7%

33.3%

Q. Grimes

+ Made

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

V. Edgecombe

+ Made

Catch Shoot

75.0%

203.4/562.5

35.7%

37.0%

Q. Grimes

+ Made

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

Q. Grimes

X Miss

Stepback

75.7%

829.0/2255.1

36.5%

34.2%

T. Maxey

X Miss

Stepback

87.9%

911.9/2405.3

37.9%

35.8%

T. Maxey

+ Made

Pullup

87.9%

911.9/2405.3

37.9%

35.8%

Q. Grimes

X Miss

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

C. Payne

+ Made

Pullup

90.7%

704.1/1939.0

36.7%

34.7%

K. Oubre Jr.

X Miss

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

K. Oubre Jr.

+ Made

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

C. Payne

X Miss

Pullup

90.7%

704.1/1939.0

36.7%

34.7%

T. Maxey

X Miss

Stepback

87.9%

911.9/2405.3

37.9%

35.8%

V. Edgecombe

+ Made

Catch Shoot

75.0%

203.4/562.5

35.7%

37.0%

V. Edgecombe

+ Made

Catch Shoot

75.0%

203.4/562.5

35.7%

37.0%

K. Oubre Jr.

X Miss

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

K. Oubre Jr.

X Miss

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

Q. Grimes

+ Made

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

V. Edgecombe

+ Made

Pullup

75.0%

203.4/562.5

35.7%

33.3%

J. Walker

X Miss

Catch Shoot

69.0%

122.3/396.3

32.5%

33.6%

Q. Grimes

X Miss

Stepback

75.7%

829.0/2255.1

36.5%

34.2%

C. Payne

X Miss

Pullup

90.7%

704.1/1939.0

36.7%

34.7%

T. Maxey

+ Made

Pullup

87.9%

911.9/2405.3

37.9%

35.8%

V. Edgecombe

+ Made

Stepback

75.0%

203.4/562.5

35.7%

33.3%

Q. Grimes

+ Made

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

K. Oubre Jr.

X Miss

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

K. Oubre Jr.

+ Made

Catch Shoot

75.1%

1149.0/3502.0

33.1%

34.4%

T. Maxey

+ Made

Pullup

87.9%

911.9/2405.3

37.9%

35.8%

Q. Grimes

+ Made

Catch Shoot

75.7%

829.0/2255.1

36.5%

37.9%

J. Edwards

+ Made

Catch Shoot

69.6%

140.2/400.4

34.6%

35.8%

J. Edwards

X Miss

Running

69.6%

140.2/400.4

34.6%

32.3%

D. Terry

+ Made

Pullup

71.0%

119.2/343.7

34.5%

32.2%

TOTALS

12.9 exp

Actual: 21 made

+8.1 luck

PHI Player Summary

Player	FT%	Shots	Made	Expected Makes	Luck	Point Impact
V. Edgecombe	75.0%	7	6	2.47	+3.53	+9.5
Q. Grimes	75.7%	8	5	2.96	+2.04	+5.5
T. Maxey	87.9%	7	4	2.56	+1.44	+3.9
D. Terry	71.0%	1	1	0.32	+0.68	+1.8
K. Oubre Jr.	75.1%	7	3	2.41	+0.59	+1.6
J. Edwards	69.6%	2	1	0.68	+0.32	+0.9
C. Payne	90.7%	3	1	1.04	-0.04	-0.1
D. Barlow	63.6%	1	0	0.29	-0.29	-0.8
J. Walker	69.0%	1	0	0.34	-0.34	-0.9
TOTAL		37	21	13.07	+7.93	+21.4

Player

FT%

Shots

Made

Expected Makes

Luck

Point Impact

V. Edgecombe

75.0%

2.47

+3.53

+9.5

Q. Grimes

75.7%

2.96

+2.04

+5.5

T. Maxey

87.9%

2.56

+1.44

+3.9

D. Terry

71.0%

0.32

+0.68

+1.8

K. Oubre Jr.

75.1%

2.41

+0.59

+1.6

J. Edwards

69.6%

0.68

+0.32

+0.9

C. Payne

90.7%

1.04

-0.04

-0.1

D. Barlow

63.6%

0.29

-0.29

-0.8

J. Walker

69.0%

0.34

-0.34

-0.9

TOTAL

13.07

+7.93

+21.4

Final Calculation

MIN

Actual 3PM: 10
Expected 3PM: 12.1
Luck (makes above/below expected): -2.1

PHI

Actual 3PM: 21
Expected 3PM: 13.1
Luck (makes above/below expected): +7.9

ORB Adjustment Calculation

Each lucky make isn't worth a full 3 points because missed 3s generate offensive rebounds (~27% ORB rate) that lead to additional scoring (~1.1 points per possession). We subtract this "lost opportunity" value:

Point Adjustment = Raw 3PT Value - ORB Opportunity Cost = (3 x Luck) - (ORB Rate x PPP x |Luck|) = (3 x Luck) - (0.27 x 1.1 x |Luck|) = 2.7 x Luck

Team	Luck	Raw 3PT Value	ORB Adjustment	Net Point Impact
MIN	-2.14	-6.4	+0.6	-5.8
PHI	+7.93	+23.8	-2.4	+21.4

Net Result

The luck differential of 10.1 makes in favor of PHI translates to 27.2 points of margin swing after ORB adjustment.

Actual margin: MIN -27

Adjusted margin: MIN +0.2

Why FT% as a Prior?

Free throw shooting is a strong predictor of three-point ability because both require:

Consistent shooting mechanics
Touch and feel for the ball
Mental focus under pressure

Our calibration across 10 seasons (3,073 player-seasons) found:

R-squared = 0.11 for FT% predicting overall 3P%
A player with 90% FT expects ~38% 3P prior
A player with 70% FT expects ~34% 3P prior

This anchors expectations appropriately: we don't expect poor free throw shooters to suddenly become elite 3PT shooters, even if they've had a hot streak.