This is a graded in-class assignment. Show all your work in R Markdown files. Submit compiled HTML files only.

Question 1

Suppose drawing two cards from a deck and tossing two coins. Answer the following questions.

  1. What is the experiment?

    The experiment is “drawing two cards from a deck and tossing two coins”.

  2. What is “getting two heads and two aces or one head one tail one queen one king”? Pick one (Event / Outcome / Sample Space)

    Event.

  3. Give an example of two mutually exclusive events.

    Event A: Jack of Spades / Queen of Hearts / Heads / Heads Event B: Ace of Clubs / Ace of Diamonds / Tails / Tails

  4. What is the probability of getting one head and one tail (in any order) and getting two Jacks?

    #It can be either HT or TH so probability of the coins is 1/2
#Getting the first Jack has probability of 4/52
#Getting the second Jack has probability of 3/51
1/2*4/52*3/51
    ## [1] 0.002262443

  5. How many different outcomes can there be? Assume ordering is important (e.g. HT and TH are different outcomes).

    #Two outcomes per coin
#52 outcomes for the first card draw
#51 outcomes for the second card draw
#Multiplication rule
2*2*52*51
    ## [1] 10608

Question 2

In how many ways can you arrange the letters of “COMPUTERLAB”?

  1. Any order.

    #Permutation rule
#11 characters no repetitive letters
factorial(11)
    ## [1] 39916800

  2. Vowels together?

    #4 vowels, 7 consonants
#Assume all vowels are a single "letter". So 8 characters.
#But vowels permutate within the single "letter".
#Multiplication rule
factorial(7+1)*factorial(4)
    ## [1] 967680

  3. Vowels in alphabetical order?

    #We start with all the permutations 11!
#For any permutation there can be only one ordering of vowels.
#For instance COMPUTERLAB is not valid but CAMPETORLUB is valid
#So remove invalid permutations with division
factorial(11)/factorial(4)
    ## [1] 1663200

  4. There should be no consecutive vowels?

    #There are 7 consonants, 4 vowels.
# Assume Xs are consonants and .s are potent vowel places.
# .X.X.X.X.X.X.X.
#Consonants can permutate in any order so 7! there
#8 places for vowels but only 4 vowels.
# So it is a permutation of 4 out of 8 places.
factorial(7)*(factorial(8)/factorial(8-4))
    ## [1] 8467200

Question 3

In how many ways can you arrange the letters of “HETEROSKEDASTICITY”?

# 18 characters.
# 7 vowels, 11 consonants
# 3 Es, 3 Ts, 2 Is, 2 Ss

  1. Any order.

    #By the formula of permutation with repetitive letters
#Assign the value to all_perms object
all_perms<-factorial(18)/(factorial(3)*factorial(3)*factorial(2)*factorial(2))
all_perms
    ## [1] 4.446093e+13

  2. Vowels together?

    #Assume all vowels are single "character" again. So 12 characters
(factorial(11+1)/(factorial(3)*factorial(2)))*(factorial(7)/(factorial(3)*factorial(2)))
    ## [1] 16765056000

  3. Vowels in alphabetical order?

    #Same as the last question. But be careful about identical vowels.
all_perms/(factorial(7)/(factorial(3)*factorial(2)))
    ## [1] 105859353600

  4. There should be no consecutive vowels?

    #Same as the last question. But be careful about identical vowels.

(factorial(11)/factorial(3)*factorial(2))*(factorial(12)/factorial(12-7))/(factorial(3)*factorial(2))
    ## [1] 4.425975e+12

Question 4

Suppose you are putting the top 16 football teams in 4 groups evenly (each group should consist of 4 teams). In how many different ways can you arrange the teams?

#It is either a chain of combinations or just grouping combination
choose(16,4)*choose(12,4)*choose(8,4)
## [1] 63063000

Question 5

There are 20 people; 10 from Ankara, 10 from Istanbul.

  1. Suppose you want to form a group of 5 people with at least 1 person from Ankara and Istanbul. In how many ways can you form such a group?

    #Calculate as if no rules. It is the combination of 20 to 5.
#Then remove the combinations of all Ankara or all Istanbul people
choose(20,5) - choose(10,5) - choose(10,5)
    ## [1] 15000

  2. In how many ways can you form a group of 3 people from Istanbul and 4 people from Ankara?

    #Simply separate combinations with multiplication rule.
choose(10,3)*choose(10,4)
    ## [1] 25200