NFL Preseason (49ers @ Broncos): How many TOUCHDOWNS will be SCORED in the 1st Half?

Inputs To Solve

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

SF_TDperG = 2.2
Den_TDperG = 2.4

total_TD = SF_TDperG + Den_TDperG

print("The total expected TOUCHDOWNS for both teams per game is is %s." % (round(total_TD,3)))

The total expected TOUCHDOWNS for both teams per game is is 4.6.

## Inputs Defined in the Problem

period_of_game = .5
TDs = [0,1,2]

Method to Solve

[1] Estimate lambda_td - expected number of a TOUCHDOWNS scored over one half for both teams.
[2] Use the Poisson Distribution to compute the probability of 0, 1 or 2 TOUCHDOWNS (p_2orless) being scored in the 1st Half.

## [1]

lambda_td = total_TD * period_of_game
print("lambda_td ~ the total expected TOUCHDOWNS over one half by the both teams")
print("lambda_td ~ %s * %s" % (round(total_TD,3),period_of_game))
print("lambda_td ~ %s" % (round(lambda_td,3)))

lambda_td ~ the total expected TOUCHDOWNS over one half by the both teams
lambda_td ~ 4.6 * 0.5
lambda_td ~ 2.3

## [2]
import math
str_ = ""
print("The probability of k events occurring in a Poisson interval = e^(-lambda) * (lambda^k)/k!")
print("where k = [0,1,2]")
print(' ')

p_2orless = 0
for i in range(TDs[2]+1):
    p_2orless += math.exp(-lambda_td)*(lambda_td**(i))/(math.factorial(i))
    
    if(i<TDs[2]):
        str_ += str(round(math.exp(-lambda_td)*(lambda_td**(i))/(math.factorial(i)),3)) + " + "
        print("e^(-%s) * (%s^%s)/%s! + " % (round(lambda_td,2),round(lambda_td,2),(i),(i)))
    else:
        str_ += str(round(math.exp(-lambda_td)*(lambda_td**(i))/(math.factorial(i)),3))
        print("e^(-%s) * (%s^%s)/%s!" % (round(lambda_td,2),round(lambda_td,2),(i),(i)))

print(' ')
print(str_)

The probability of k events occurring in a Poisson interval = e^(-lambda) * (lambda^k)/k!
where k = [0,1,2]
 
e^(-2.3) * (2.3^0)/0! + 
e^(-2.3) * (2.3^1)/1! + 
e^(-2.3) * (2.3^2)/2!
 
0.1 + 0.231 + 0.265

Solution

print("The probability 2 or Fewer TOUCHDOWNS are scored in the 1st Half is ~%s" % round(p_2orless,3))

The probability 2 or Fewer TOUCHDOWNS are scored in the 1st Half is ~0.596

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 8/19/2019