NCF (North Carolina @ Wake Forest): Which will OCCUR FIRST in the 1st Quarter?

6:05 PM
Passing Touchdown
Rushing Tuchdown or Neither occur

Inputs To Solve

North Carolina Touchdown Stats
Wake Forest Touchdown Stats

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

passing_TDs = 1.3 + 1.9
rushing_TDs = 1.7 + 1.6

print('The total number of observed Passing TOUCHDOWNS per Game for both teams is %s' % passing_TDs)
print('The total number of observed Rushing TOUCHDOWNS per Game for both teams is %s' % rushing_TDs)

The total number of observed Passing TOUCHDOWNS per Game for both teams is 3.2
The total number of observed Rushing TOUCHDOWNS per Game for both teams is 3.3

## Inputs Defined in the Problem

period_of_game = .25
neither_happens = 0
first_to = 1

Method to Solve

[1] Estimate lambda_pass - expected arrival rate of a Passing TD per Quarter
[2] Estimate lambda_rush - expected arrival rate of a Rushing TD per Quarter
[3] Use the Poisson Distribution to compute the probability neither a Passing or Rushing TD occur (p_neither).
[4] Use the Poisson Distribution assumptions and this equation to compute the probability of a Rushing TD occuring before a Passing TD (p_rush).
[5] Compute the probability a Rushing TD occurs before a passing TD including the probability neither occur. P_RUSH

## [1]

lambda_pass = passing_TDs * period_of_game
print("lambda_pass ~ the total expected Passingn TD per Quarter")
print("lambda_pass ~ %s * %s" % (round(passing_TDs,3),round(period_of_game,3)))
print("lambda_pass ~ %s" % (round(lambda_pass,3)))

lambda_pass ~ the total expected Passingn TD per Quarter
lambda_pass ~ 3.2 * 0.25
lambda_pass ~ 0.8

## [2]

lambda_rush = rushing_TDs * period_of_game
print("lambda_rush ~ the total expected Passingn TD per Quarter")
print("lambda_rush ~ %s * %s" % (round(rushing_TDs,3),round(period_of_game,3)))
print("lambda_rush ~ %s" % (round(lambda_rush,3)))

lambda_rush ~ the total expected Passingn TD per Quarter
lambda_rush ~ 3.3 * 0.25
lambda_rush ~ 0.825

## [3]

import math

p_p = math.exp(-lambda_pass )*(lambda_pass **neither_happens)/(math.factorial(neither_happens))
p_r = math.exp(-lambda_rush )*(lambda_rush **neither_happens)/(math.factorial(neither_happens))

p_neither = p_p * p_r

print('The probability that neither a Passing or Rushing TD occur in the 3rd Quarter is %s' % round(p_neither,3))

The probability that neither a Passing or Rushing TD occur in the 3rd Quarter is 0.197

## [4]

n = first_to
m = first_to

p_rush = 0
for k in range(n,n+m):
    ans = math.factorial(n+m-1)/(math.factorial((n+m-1)-k)*math.factorial(k))
    ans1 = (lambda_rush/(lambda_rush+lambda_pass))**k
    ans2 = (lambda_pass/(lambda_rush+lambda_pass))**(n+m-1-k)
    p_rush += (ans*ans1*ans2)
    
print('The probability a Rushing TD occurs before a Passing TD in the 3rd Quarter is %s' % round(p_rush,3))

The probability a Rushing TD occurs before a Passing TD in the 3rd Quarter is 0.508

## [5]

P_RUSH = p_neither + p_rush*(1-p_neither)

Solution

print("The probability a Rushing Touchdown occurs before a Passing Touchdown or neither occur in the 3rd Quarter is ~%s" % round(P_RUSH,3))

The probability a Rushing Touchdown occurs before a Passing Touchdown or neither occur in the 3rd Quarter is ~0.605

Info

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import sys
print(sys.version)

3.6.5 |Anaconda, Inc.| (default, Mar 29 2018, 13:32:41) [MSC v.1900 64 bit (AMD64)]

Posted on 9/13/2019