PGA BMW Championship/ATP Cincinnati: Which side will record a HIGHER TOTAL?

10:42AM
D. Schwartzman v. R. Gasquet: 1st Set Aces or Tie
J. Furyk + P. Mickelson: Score on Hole 2

Inputs Needed To Solve

2019 Aces Per Service Games Played
Schwartzman
Gasquet

Medinah CC Course Stats

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

# atp
schwartzman_servwon = .73
gasquet_servwon = .79

Schwartzman_ACEperSERVE = 95/559
Gasquet_ACEperSERVE = 129/260

# pga (estimate given the course has not been played yet)
hole_2 = {'2':50,
          '3':313,
          '4':44,
          '5':1}

print("The ratio of Aces per Serves for Schwartzman is %s" % round(Schwartzman_ACEperSERVE,3))
print("The ratio of Aces per Serves for Gasquet is %s" % round(Gasquet_ACEperSERVE,3))

The ratio of Aces per Serves for Schwartzman is 0.17
The ratio of Aces per Serves for Gasquet is 0.496

Method To Solve

[1] Use Monte Carlo Simulation and simulate 9999 Sets between Schwartzman and Gasquet using their respective Service Games Won Percent to obtain an estimate for the number of times each player will serve and therefore an estimate of the number of Aces.
[2] Create a distribution of Aces observed by simulating one set between the 2 players. aces_dist
[3] Enumerate all the possible combinations of observed scores and their respective probabilities for 2 golfers on the 2nd hole. hole_2_dist
[4] Enumerate all the possible combinations of Aces and Hole 2 Scores.
[5] The probability that Furyk and Mickelson’s score is greater than 1st Aces (p_golfers_higher) is the sum of the probabilities of the respective outcomes.

## [1]

import numpy as np
import pandas as pd

def sim_game(server,player1,player2):
    if server:
        serve = np.random.choice([1,0],1,p=[player1,1-player1])
        if serve == 1:
            ans = '1'
        else:
            ans = '0'
        
    else:
        serve = np.random.choice([1,0],1,p=[player2,1-player2])
        if serve == 1:
            ans = '0'
        else:
            ans = '1'
        
    return ans

def set_over(winner):
    set_over = False
    
    if winner.count('1') >= 7:
        set_over = True
    if winner.count('0') >= 7:
        set_over = True
    if winner.count('1') >=6 and winner.count('0') < 5:
        set_over = True
    if winner.count('0') >=6 and winner.count('1') < 5:
        set_over = True
    if winner.count('0') == 6 and winner.count('1') == 6:
        set_over = True
    
    return set_over

def sim_set(player1,player2):
    winner = ''
    server = True
    set_over_ = False
    aces = 0

    while not set_over_:
        game = sim_game(server,player1,player2)
        winner += str(game)
        
        if server:
            a = np.random.choice([1,0],1,p=[Schwartzman_ACEperSERVE,1-Schwartzman_ACEperSERVE])
        else:
            a = np.random.choice([1,0],1,p=[Gasquet_ACEperSERVE,1-Gasquet_ACEperSERVE])

        aces += a
        
        server = not server
        set_over_ = set_over(winner)
    
    return aces

iterations = 9999
aces = []

for i in range(iterations):
    aces.append(int(sim_set(gasquet_servwon,schwartzman_servwon)))

## [2]

aces_dist = {}

for a in aces:
    if a in aces_dist:
        aces_dist[a] += 1
    else:
        aces_dist[a] = 1
        
t=sum(aces_dist.values())
for i in aces_dist:
    aces_dist[i] = aces_dist[i]/t
    
aces_dist

{6: 0.0502050205020502,
 4: 0.21722172217221722,
 2: 0.2078207820782078,
 3: 0.2611261126112611,
 7: 0.0178017801780178,
 5: 0.12361236123612361,
 1: 0.0962096209620962,
 0: 0.0223022302230223,
 8: 0.0033003300330033004,
 9: 0.00040004000400040005}

## [3]

t=sum(hole_2.values())
for i in hole_2:
    hole_2[i] = hole_2[i]/t

y = np.array([(a,b) for a in hole_2.keys()for b in hole_2.keys()])
y = y.astype(int)
scores = pd.DataFrame(y)
scores['total_strokes'] = scores.sum(axis=1)
scores.head()

	0	1	total_strokes
0	2	2	4
1	2	3	5
2	2	4	6
3	2	5	7
4	3	2	5

z = np.array([(a,b) for a in hole_2.values() for b in hole_2.values()])
probability = pd.DataFrame(z)
probability['p'] = probability.product(axis=1)
probability.head()

	0	1	p
0	0.122549	0.122549	0.015018
1	0.122549	0.767157	0.094014
2	0.122549	0.107843	0.013216
3	0.122549	0.002451	0.000300
4	0.767157	0.122549	0.094014

hole_2_dist = {}
for s in set(scores['total_strokes']):
    hole_2_dist[s] = probability['p'][scores['total_strokes']==s].sum()

hole_2_dist

{4: 0.01501826220684352,
 5: 0.1880286428296809,
 6: 0.6149617935409457,
 7: 0.16606593617839296,
 8: 0.015390715109573244,
 9: 0.000528642829680892,
 10: 6.007304882737409e-06}

## [4]

y = np.array([(a,b) for a in hole_2_dist.keys()for b in aces_dist.keys()])
y = y.astype(int)
scores = pd.DataFrame(y)
scores['golfers_higher'] = scores[0] > scores[1]
scores.head()

	0	1	golfers_higher
0	4	6	False
1	4	4	False
2	4	2	True
3	4	3	True
4	4	7	False

z = np.array([(a,b) for a in hole_2_dist.values() for b in aces_dist.values()])
probability = pd.DataFrame(z)
probability['p'] = probability.product(axis=1)
probability.head()

	0	1	p
0	0.015018	0.050205	0.000754
1	0.015018	0.217222	0.003262
2	0.015018	0.207821	0.003121
3	0.015018	0.261126	0.003922
4	0.015018	0.017802	0.000267

## [5]

p_golf_higher = probability['p'][scores['golfers_higher']==True].sum()

Solution

print("The probability Furyk and Mickelson's score is greater than the 1st Set Aces is ~%s" % round(p_golf_higher,3))

The probability Furyk and Mickelson's score is greater than the 1st Set Aces is ~0.909

Info

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Posted on 8/15/2019