1. Please explain Bayes Theorem in your own words, and give an example. Less than 10 sentences. Also, write out the formula. Pick up on how to to type equations in R Markdown using Latex terminology

Bayes Theorem: Bayes Theorem is when we update the probability based on new information or evidence. By being able to use updated information in our formula, it can help us create a more accurate probability hypothesis. This theorem is an extension of conditional probabilities. Since we know conditional probability is what we use to predict the probability of A|B (A happening given that B happened). By using Bayes Theorem, we can essentially calculate the probability of A occurring if we know the probability of another event related to B occurring.

The formula is listed below.

\(P(A \mid B)\) = \(\displaystyle \frac{P(B\mid A)* P(A) }{P(B)}\)

2 Bayes’ Theorem can yield surprising results. Take a look at Open Stats textbook in Dropbox folder (Dropbox link on Course Introduction and Materials) Guided Practice 3.43 (page 107, also attached below) and attempt to solve this using Bayes’ formula. Interpret the results. Then use the attached code and solve via a tree diagram in R. You will need to change the initial parameters to the appropriate values. Attach your final graph to your submission, and your code.

rm(list=ls()) # empty environment
sessionInfo()
## R version 4.3.3 (2024-02-29 ucrt)
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## time zone: America/New_York
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## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
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## loaded via a namespace (and not attached):
##  [1] digest_0.6.35     R6_2.5.1          fastmap_1.1.1     xfun_0.42        
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##  [9] lifecycle_1.0.4   cli_3.6.2         sass_0.4.8        jquerylib_0.1.4  
## [13] compiler_4.3.3    rstudioapi_0.15.0 tools_4.3.3       evaluate_0.23    
## [17] bslib_0.6.1       yaml_2.3.8        rlang_1.1.3       jsonlite_1.8.8
library(BiocManager)
BiocManager::install("Rgraphviz")
## Bioconductor version 3.18 (BiocManager 1.30.22), R 4.3.3 (2024-02-29 ucrt)
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## Old packages: 'bslib', 'pkgbuild', 'processx', 'psych', 'sass'
require("Rgraphviz")
## Loading required package: Rgraphviz
## Loading required package: graph
## Loading required package: BiocGenerics
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## Attaching package: 'BiocGenerics'
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BiocManager::valid()
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## [1] Rgraphviz_2.46.0    graph_1.80.0        BiocGenerics_0.48.1
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##  [1] digest_0.6.35     R6_2.5.1          fastmap_1.1.1     xfun_0.42        
##  [5] cachem_1.0.8      knitr_1.45        htmltools_0.5.7   rmarkdown_2.26   
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2.1 QUESTION: Guided Practice 3.43: Jose visits campus every Thursday evening. However, some days the parking garage is full, often due to college events. There are academic events on 35% of evenings, sporting events on 20% of evenings, and no events on 45% of evenings. When there is an academic event, the garage fills up about 25% of the time, and it fills up 70% of evenings with sporting events. On evenings when there are no events, it only fills up about 5% of the time. If Jose comes to campus and finds the garage full, what is the probability that there is a sporting event? Use a tree diagram to solve this problem.

Below, I have listed the probabilities for each type of event along with their corresponding % of how full the garage gets.

Academic Events: 35%

Sporting Events: 20%

No events: 45%

# Sporting Event
PA1 <- .20 

# Academic Event
PA2 <- .35  

# No Event
PA3 <- .45  

# Garage Full - Sporting Event
PB_A1 <- .70  

# Garage Full - Academic Event
PB_A2 <- .25 

# Garage Full - No Event
PB_A3 <- .05 

PA1_B <- (PA1 * PB_A1) / ((PA1*PB_A1)+(PA2*PB_A2)+(PA3*PB_A3))
PA1_B 
## [1] 0.56

As we can see above, the probability of there being a sporting event when Jose sees the garage full is 56%.

Our numerator was probability of the garage being full during a sporting event * the probability of there being a sporting event

\(P(B \mid A)* P(A)\) = 0.20*0.70 = 0.14

We then set up our denominator being the sum of each events probability * the probability of the garage being full for that event

\({P(B)}\) = (0.20 * 0.70)+(0.35 * 0.25)+(0.45 * 0.05)

(0.14+0.0875+0.0225) = 0.25

\(P(A \mid B)\) = 0.14/0.025 = 0.56 = 56%

PC1 <- 1-PB_A1 # Not Full - Sport
PC2 <- 1-PB_A2 # Not Full - Academic
PC3 <- 1-PB_A3 # Not Full - No Event

# Create Joint Probabilities #

PAB1 <- PA1 * PB_A1
PAC1 <- PA1 * PC1

PAB2 <- PA2 * PB_A2
PAC2 <- PA2 * PC2

PAB3 <- PA3 * PB_A3
PAC3 <- PA3 * PC3

Graph

Nodes

node1<-"P"
node2<-"Sport"
node3<-"Academic"
node4<-"No Event"
node5<-"Sport Full"
node6<-"Sport N_Full"
node7<-"Acad Full"
node8 <-"Acad N_Full"
node9 <- "No Event Full"
node10 <-"No Event N_Full"
nodeNames<-c(node1,node2,node3,node4,node5,node6,node7,node8,node9,node10)

rEG <- new("graphNEL", 
           nodes=nodeNames, 
           edgemode="directed"
)

Branches

rEG <- addEdge(nodeNames[1], nodeNames[2], rEG, 1)
rEG <- addEdge(nodeNames[1], nodeNames[3], rEG, 1)
rEG <- addEdge(nodeNames[1], nodeNames[4], rEG, 1)
rEG <- addEdge(nodeNames[2], nodeNames[5], rEG, 1)
rEG <- addEdge(nodeNames[2], nodeNames[6], rEG, 1)
rEG <- addEdge(nodeNames[3], nodeNames[7], rEG, 1)
rEG <- addEdge(nodeNames[3], nodeNames[8], rEG, 1)
rEG <- addEdge(nodeNames[4], nodeNames[9], rEG, 1)
rEG <- addEdge(nodeNames[4], nodeNames[10], rEG, 1)

eAttrs <- list()
 
q<-edgeNames(rEG)

Adding Probability

eAttrs$label <- c(toString(PA1),
                  toString(PA2),
                  toString(PA3), 
                  toString(PB_A1),
                  toString(PC1), 
                  toString(PB_A2), 
                  toString(PC2), 
                  toString(PB_A3), 
                  toString(PC3)
                  )
names(eAttrs$label) <- c(q[1],q[2], q[3], q[4], q[5], q[6], q[7], q[8], q[9])
edgeAttrs<-eAttrs

# Set the color, etc, of the tree
attributes<-list(node=list(label="example", 
                           fillcolor="lightblue", 
                           fontsize=""),
 edge=list(color="black"),
 graph=list(rankdir="LR"))
 
#Plot the probability tree using Rgraphvis
plot(rEG, edgeAttrs=eAttrs, attrs=attributes)

nodes(rEG)
##  [1] "P"               "Sport"           "Academic"        "No Event"       
##  [5] "Sport Full"      "Sport N_Full"    "Acad Full"       "Acad N_Full"    
##  [9] "No Event Full"   "No Event N_Full"