1 Introduction

This blog post is aimed at answering a series of questions related to the World Series using the rules of probability and discrete probability functions to answer the questions. The target person of this blog post is a HR manager tasked with hiring a data scientist.

2 Solutions

library(ggplot2)
library(tidyverse)

Question 1: The probability for Braves win the World Series should be: $$P=P(Win~in~4~games)+P(Win~in~5~games)+P(Win~in~6~games)+P(Win~in~7~games)$$

Prob_BravesWinIn7=choose(6,3)*0.55^3*0.45^3*0.55+choose(5,3)*0.55^3*0.45^2*0.55+choose(4,3)*0.55^3*0.45*0.55+0.55^4

Question2: The probability of Brave wins WS should be: $$P=P(Win~in~4~games)+P(Win~in~5~games)+P(Win~in~6~games)+P(Win~in~7~games)$$

PB = seq(.1,1,.05)
Prob=c()
for(prob in PB){
  Prob<-append(Prob,dnbinom(0,4,prob)+dnbinom(1,4,prob)+dnbinom(2,4,prob)+dnbinom(3,4,prob))}
tibble_1<-tibble(PB,Prob)
ggplot(tibble_1,aes(x=PB,y=Prob))+geom_point()+geom_line()+ylab("Probability of Braves wins WS")+xlab('Probability of Braves wins each game')

Question 3:

num_games = seq(7,151, by=2)
Pb= 0.55
num_wins=ceiling(num_games/2)
num_loss=num_games-num_wins
P_B_WS=pnbinom(num_loss,num_wins,Pb)
plot(num_games,P_B_WS)+abline(h=.8,col="red")

## integer(0)

num_games[which(P_B_WS>=.8)][1]

## [1] 71

Question 4:

Pb= 0.6

num_games = seq(7,151, by=2)
num_wins=ceiling(num_games/2)
num_loss=num_games-num_wins

P_B_WS=pnbinom(num_loss,num_wins,Pb)
plot(num_games,P_B_WS)+abline(h=.8,col="red")

## integer(0)

num_games[which(P_B_WS>=.8)]

##  [1]  17  19  21  23  25  27  29  31  33  35  37  39  41  43  45  47  49  51  53
## [20]  55  57  59  61  63  65  67  69  71  73  75  77  79  81  83  85  87  89  91
## [39]  93  95  97  99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129
## [58] 131 133 135 137 139 141 143 145 147 149 151

P_B_WS_SEQ=c()
seq_Pb=seq(.5,1,by=.01)
for(prob in seq_Pb){
  P_B_WS=pnbinom(num_loss,num_wins,prob)
  P_B_WS_SEQ<-append(P_B_WS_SEQ,num_games[which(P_B_WS>=.8)][1])
}
plot(x=seq_Pb,y=P_B_WS_SEQ)+abline(h=7,col="red")

## integer(0)

# We could not see clearly when Probability >.65, let's limit y axis to see more clearly
plot(x=seq_Pb,y=P_B_WS_SEQ,ylim=c(7,10))+abline(h=7,col="red")

## integer(0)

We could see that when probability is greater than .65, the shortest length is 7 for Braves has 80% to win.

Question 5:

The question could be interpreted to: P(Pb=0.55| Braves win WS in 7 games)= P(Braves win WS in 7 games|Pb=0.55)/ P(Braves win WS in 7 games) or P(Pb=0.45| Braves win WS in 7 games)= P(Braves win WS in 7 games|Pb=0.45)/ P(Braves win WS in 7 games) or

# Pb=0.55
(choose(6,3)*.55^3*.45^3*.55/(choose(6,3)*.55^3*.45^3*.55+choose(6,3)*.45^3*.55^3*.45))

## [1] 0.55

(choose(6,3)*.45^3*.55^3*.45/(choose(6,3)*.55^3*.45^3*.55+choose(6,3)*.45^3*.55^3*.45))

## [1] 0.45

The result shows that the results of the probabilty for Pb =.55 and .45 are the probability themselves. This is might because Braves win WS in 7 games are independant with what’s the probability should be. So P(Braves win WS in 7 games|Pb=.45/.55 would equal P(Braves win WS in 7 games)*P(Pb=.45/.55)