MLB (Phillies @ Giants): How many EXTRA-BASE HITS will be recorded during Innings 5-8?

Inputs To Solve

XBH per Game by Team (Phillies and Giants)

##### User Estimates #####
PHI_XperG = (215+17+149)/118
SFG_XperG = (236+21+126)/119

expected_total_Xs = PHI_XperG + SFG_XperG

print("Use %s + %s = %s as total expected XBH be recorded in the game." % (round(PHI_XperG,3),round(SFG_XperG,3),round(expected_total_Xs,3)))

Use 3.229 + 3.218 = 6.447 as total expected XBH be recorded in the game.

## Inputs Defined in the Problem
period_of_innings = 4
XBH = [0,1,2,3]

Method to Solve

[1] Estimate lambda - expected arrival rate of XBH for 3 Innings (lambda_)
[2] Use the Poisson Distribution to compute the probability of 0, 1, 2 or 3 XBH being recorded during Innings 5-8 (p_3orless)

## [1]

lambda_ = expected_total_Xs * period_of_innings / 9
print("lambda = the total expected RUNS SCORED over %s innings" % period_of_innings)
print("lambda = %s * %s / %s" % (round(expected_total_Xs,3),period_of_innings,9))
print("lambda ~ %s" % (round(lambda_,3)))

lambda = the total expected RUNS SCORED over 4 innings
lambda = 6.447 * 4 / 9
lambda ~ 2.865

## [2]

import math

str_ = ""
print("The probability of k events occurring in a Poisson interval = e^(-lambda) * (lambda^k)/k!")
print("where k = [0,1,2,3]")
print(' ')

p_3orless = 0
for i in XBH:
    p_3orless += math.exp(-lambda_)*(lambda_**(i))/(math.factorial(i))
    
    if(i<=2):
        str_ += str(round(math.exp(-lambda_)*(lambda_**(i))/(math.factorial(i)),3)) + " + "
        print("e^(-%s) * (%s^%s)/%s! + " % (round(lambda_,2),round(lambda_,2),(i),(i)))
    else:
        str_ += str(round(math.exp(-lambda_)*(lambda_**(i))/(math.factorial(i)),3))
        print("e^(-%s) * (%s^%s)/%s!" % (round(lambda_,2),round(lambda_,2),(i),(i)))

print('')
print('p_3orless = ' + str_)

The probability of k events occurring in a Poisson interval = e^(-lambda) * (lambda^k)/k!
where k = [0,1,2,3]

e^(-2.87) * (2.87^0)/0! +
e^(-2.87) * (2.87^1)/1! +
e^(-2.87) * (2.87^2)/2! +
e^(-2.87) * (2.87^3)/3!

p_3orless = 0.057 + 0.163 + 0.234 + 0.223

Solution

print("The probability that 3 XBH or Fewer are recorded during Innings 5-8 is ~%s" % round(p_3orless,3))

The probability that 3 XBH or Fewer are recorded during Innings 5-8 is ~0.677

Posted on 8/12/2019