NFL Hall of Fame Game (Broncos v. Falcons): Will a TOUCHDOWN be SCORED in BOTH the 1st and 2nd Quarter?

8:10PM
Yes: TD in both the 1st and 2nd Qtr
No: No TD in at least 1 of those Qtrs

Inputs Needed to Solve

TDs per Game by team 2018 (Broncos & Falcons)

##### User Estimates #####
DEN_TDperG = 2.4
ATL_TDperG = 3.1

total_TDs = DEN_TDperG + ATL_TDperG

print("The total expected TOUCHDOWNS for the game is %s" % round(total_TDs,3))

The total expected TOUCHDOWNS for the game is 5.5

## Inputs Defined in the Problem
period_of_time = .25
TDs = 0

Method to Solve

Estimate lambda (expected arrival rate of TDs per quarter for both teams)
Use the Poisson Distribution to compute the probability of zero TDs being scored in any one quarter
The probability that a TD is scored in both the 1st and 2nd quarter is (1 - the probability of zero TDs being scored in any one quarter) ^ 2

lambda_ = total_TDs * period_of_time
print("lambda = the total expected arrival rate of TDs scored by both teams in one quarter")
print("lambda = %s * %s" % (round(total_TDs,2),period_of_time))
print("lambda ~ %s" % (round(lambda_,2)))

lambda = the total expected arrival rate of TDs scored by both teams in one quarter
lambda = 5.5 * 0.25
lambda ~ 1.38

import math

p_zero = math.exp(-lambda_)*(lambda_**TDs)/(math.factorial(TDs))

print("The probability of k events occuring in an Poisson interval = e^(-lambda) * ((lambda^k)/k!)")
print('where k = 0 and lambda = %s' % round(lambda_,2))
print('')
print("p_zero = e^(-%s) * (%s^%s)/%s!" % (round(lambda_,2),round(lambda_,2),TDs,TDs))

The probability of k events occuring in an Poisson interval = e^(-lambda) * ((lambda^k)/k!)
where k = 0 and lambda = 1.38

p_zero = e^(-1.38) * (1.38^0)/0!

print("The probability of zero TDs being scored in any one quarter is %s" % (round(p_zero,3)))

The probability of zero TDs being scored in any one quarter is 0.253

p_final = (1-p)**2

Solution

print("The probability of a TOUCHDOWN being SCORED in BOTH the 1st and 2nd Quarter is %s" % (round(p_final,3)))

The probability of a TOUCHDOWN being SCORED in BOTH the 1st and 2nd Quarter is 0.558

Posted on 8/1/2019