IS-605 Assignment 06
Mohamed Elmoudni
October 4, 2015
PS1.1 When you roll a fair die 3 times, how many possible outcomes are there?
# Let X be the event of the rollong the die once
# Let Y be the event of the rollong the die twice
# Let Z be the event of the rollong the die three times.
X<- matrix(c(1:6))
Y<- matrix(c(1:6), nrow= 1, byrow= FALSE)
Z<- matrix(c(1:6))
# Each outcome above has 6 elements. Hence the total possible outcome is: 6*6*6
# Calculating how many outcomes by rolling the die three times using variable "count"
# Please note the possible outcome "L" is commented below.
count=0
for (i in 1:6) {
for (j in 1:6) {
for (n in 1:6) {
L <- (c(X[i], Y[j],Z[n]))
count=count+1
#print(L) #Please uncomment this line to see all possible outcomes
}
}
}
print(count)## [1] 216PS1.2 What is the probability of getting a sum total of 3 when you roll a die two times?
# Let X be the event of the rollong the die once
# Let Y be the event of the rollong the die twice
X<- matrix(c(1:6))
Y<- matrix(c(1:6), nrow= 1, byrow= FALSE)
# Finding all the outcomes using variable count.
count = 0
sum3=0
for (i in 1:6) {
for (j in 1:6){
z <- c(X[i],Y[j])
count=count+1
# find when the sum of total outcome of X and Y is 3
if (X[i] + Y[j] == 3)
sum3= sum3+1
}
}
print ("The probability of getting a sum total of 3 when you roll a die two times is:")## [1] "The probability of getting a sum total of 3 when you roll a die two times is:"(sum3/count)## [1] 0.05555556PS1.3
Assume a room of 25 strangers. What is the probability that two of them have the
same birthday? Assume that all birthdays are equally likely and equal to 1=365
each. What happens to this probability when there are 50 people in the room?
PS1.3 Solution
The first person birthday is fixed which is 1 = 365/365 .
The second person birthday either matches the first birthday, which occurs with probability 1/365,
or it does not, with probability 364/365.
The third person probability is either 2/365 when matches or 363/365 when it does not match.
We can write the probabilities of not matching 364/365 and 363/365 as 1-1/365 and 1-2/265
for persons 2 and 3 respectively .
For persons 4 and 5 as 1 - 3/365 and 1 - 4/365.
Hence, for the kth person the probability of not matching is the product of (1 - i/365) for i=1 to k-1.
And the probability of matching of is just 1 - [(1 - i / 365) for i=1 to k-1].
birthday <- function(k){
x =1
for (i in 1:k-1) {
#print(x)
x= (x * (1 - i/ 365))
#print(x)
#print(i)
}
return(1-x)
}
# Assume a room of 25 strangers. What is the probability that two of them have the same birthday
birthday(25)## [1] 0.5686997# Assume a room of 50 strangers. What is the probability that two of them have the same birthday
birthday(50)## [1] 0.9703736Problem Set 2
Write a program to take a document in English and print out the estimated probabilities
for each of the words that occur in that document. Your program should take in a le
containing a large document and write out the probabilities of each of the words that
appear in that document. Please remove all punctuation (quotes, commas, hyphens etc)
and convert the words to lower case before you perform your calculations.
# Load the text file
datapath <- "C:/CUNY/Courses/IS605/Assignments/Assignment06/assign6.sample.txt"
text <- scan(datapath, what = 'char', sep = '\n')
# Convert to lower case
text <- tolower(text)
# remove all punctuation (quotes, commas, hyphens etc)
text.list <- unlist(strsplit(text, "[^a-z]+"))
# Remove empty spaces
text.list<- text.list[text.list != ""]
# Compute N, the number of tokens
N<- length(text.list)
# Convert to vector
text.vector<-unlist(text.list)
freq.list<-table(text.vector, useNA = "no")
# sort the list
sorted.freq.list<-sort(freq.list, decreasing=TRUE)
# Just curious... the top 10
top10 <- sorted.freq.list[1:10]
top10## text.vector
## the a and for in of to is said that
## 76 45 38 31 28 28 28 22 22 21# the probabilities of each of the words that appear in that document
data.frame(word=names(sorted.freq.list), freq=sorted.freq.list,
probability=sorted.freq.list/N)## word freq probability
## the the 76 0.0558412932
## a a 45 0.0330639236
## and and 38 0.0279206466
## for for 31 0.0227773696
## in in 28 0.0205731080
## of of 28 0.0205731080
## to to 28 0.0205731080
## is is 22 0.0161645849
## said said 22 0.0161645849
## that that 21 0.0154298310
## it it 18 0.0132255694
## s s 16 0.0117560617
## tutwiler tutwiler 15 0.0110213079
## prison prison 13 0.0095518001
## at at 12 0.0088170463
## corrections corrections 10 0.0073475386
## are are 9 0.0066127847
## been been 9 0.0066127847
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## have have 9 0.0066127847
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## would would 1 0.0007347539
## yes yes 1 0.0007347539
## you you 1 0.0007347539Visualizing the single word frequency using wordcloud package.
library(wordcloud)## Loading required package: RColorBrewerlibrary(RColorBrewer)
dm = data.frame(word=names(sorted.freq.list), freq=sorted.freq.list)
# plot wordcloud
wordcloud(dm$word, dm$freq, random.order=FALSE, colors=brewer.pal(8, "Dark2"))
PS 2.2
Extend your program to calculate the probability of two words occurring adjacent to
each other. It should take in a document, and two words (say the and for) and compute
the probability of each of the words occurring in the document and the joint probability
of both of them occurring together. The order of the two words is not important.
Solution PS 2.2
If we affix (n ??? 1) dummy symbols (, etc.) to the start or end of a text,
then the text will contain N n-grams.
therefore, a text of length N always has N n-grams (for any n)
in our case n = 2 (bigram), then we we need 1,( 2-1) dummy symbols.
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Please note that only high frequency bigrams are being plotted.
