PGA Tour Championship - 1st Rd (Atlanta, GA): Which PAIR will record a LOWER COMBINED BACK 9 SCORE?

4:20PM
Justin Thomas + Patrick Cantlay or Tie
Brooks Koepka + Patrick Reed

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Inputs To Solve

East Lake Golf Club 2018 Course Stats

##### User Estimates #####

hole_10 = {'3':28,
           '4':79,
           '5':13}

hole_11 = {'2':10,
           '3':86,
           '4':24}

hole_12 = {'3':29,
           '4':74,
           '5':16,
           '5':1}

hole_13 = {'3':20,
           '4':86,
           '5':13,
           '6':1}

hole_14 = {'3':17,
           '4':73,
           '5':27,
           '6':3}

hole_15 = {'2':18,
           '3':76,
           '4':19,
           '5':7}

hole_16 = {'3':18,
           '4':77,
           '5':19,
           '6':6}

hole_17 = {'2':1,
           '3':28,
           '4':75,
           '5':16,
           '6':4}

hole_18 = {'3':2,
           '4':62,
           '5':50,
           '6':6}

## Inputs Defined in the Problem

golfers = ['1','2','3','4']

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Method to Solve

• [1] Enumerate all the possible combinations of observed scores and their respective probabilities for a golfer on 9 holes (pair)
• [2] Enumerate all the possible combinations of observed scores (scores) and their respective probabilities (probability) for a pair of golfers on 9 holes
• [3] The probability Brooks Koepka + Patrick Reed’s score is lower is the sum of all the probabilities in the combinations where their score is lower (p_BK_PR_lower)

## [1]

import numpy as np
import pandas as pd

## [1]

t=sum(hole_10.values())
for i in hole_10:
    hole_10[i] = hole_10[i]/t
    
t=sum(hole_11.values())
for i in hole_11:
    hole_11[i] = hole_11[i]/t
    
t=sum(hole_12.values())
for i in hole_12:
    hole_12[i] = hole_12[i]/t
    
t=sum(hole_13.values())
for i in hole_13:
    hole_13[i] = hole_13[i]/t
    
t=sum(hole_14.values())
for i in hole_14:
    hole_14[i] = hole_14[i]/t
    
t=sum(hole_15.values())
for i in hole_15:
    hole_15[i] = hole_15[i]/t
    
t=sum(hole_16.values())
for i in hole_16:
    hole_16[i] = hole_16[i]/t
    
t=sum(hole_17.values())
for i in hole_17:
    hole_17[i] = hole_17[i]/t
    
t=sum(hole_18.values())
for i in hole_18:
    hole_18[i] = hole_18[i]/t
    
y = np.array([(t,u,v,w,x,z,a,b,c) for t in hole_10.keys() for u in hole_11.keys() for v in hole_12.keys() 
              for w in hole_13.keys() for x in hole_14.keys() for z in hole_15.keys() for a in hole_16.keys()
              for b in hole_17.keys() for c in hole_18.keys()])
y = y.astype(int)
scores = pd.DataFrame(y)
scores['total_strokes'] = scores.sum(axis=1)

z = np.array([(t,u,v,w,x,z,a,b,c) for t in hole_10.values() for u in hole_11.values() for v in hole_12.values()
              for w in hole_13.values() for x in hole_14.values() for z in hole_15.values() for a in hole_16.values()
              for b in hole_17.values() for c in hole_18.values()])
probability = pd.DataFrame(z)
probability['p'] = probability.product(axis=1)

pair = {}
for s in set(scores['total_strokes']):
    pair[s] = probability['p'][scores['total_strokes']==s].sum()
    
pair

{24: 3.871561882524583e-10,
 25: 3.48701010774946e-08,
 26: 1.2427949177842029e-06,
 27: 2.3328960825676486e-05,
 28: 0.00026474145257614586,
 29: 0.001962650964754365,
 30: 0.00994404779209208,
 31: 0.03533462344702592,
 32: 0.08929012231728761,
 33: 0.16171129995247704,
 34: 0.21160121114358144,
 35: 0.2033033961230164,
 36: 0.14745682587312717,
 37: 0.08316324610279467,
 38: 0.03729879672483515,
 39: 0.013499330366800424,
 40: 0.00397390102762465,
 41: 0.0009530365215584436,
 42: 0.00018535840704149285,
 43: 2.89071618538805e-05,
 44: 3.543496246375633e-06,
 45: 3.307231429671809e-07,
 46: 2.2357504379360294e-08,
 47: 1.0061618614175353e-09,
 48: 2.5292671081376566e-11,
 49: 2.0417973416965354e-13}

## [2]

y = np.array([(a,b) for a in pair.keys() for b in pair.keys()])
y = y.astype(int)
scores = pd.DataFrame(y)
scores['BK_PR_Lower'] = scores[0] < scores[1]

scores.head()

z = np.array([(a,b) for a in pair.values() for b in pair.values()])
probability = pd.DataFrame(z)
probability['p'] = probability.product(axis=1)

probability.head()

p_BK_PR_lower = probability['p'][scores['BK_PR_Lower']==True].sum()

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Solution

print("The probability Brooks Koepka + Patrick Reed's combined score on holes 10-18 is lower is ~%s" % round(p_BK_PR_lower,3))

The probability Brooks Koepka + Patrick Reed's combined score on holes 10-18 is lower is ~0.424

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Posted on 8/22/2019