This is a graded in-class assignment with peer review. **One submission per group on paper.** Do a clean work, your style will be evaluated too. Take a snapshot of your work after peer review. Check the details of peer review guidelines on Bilgi Learn or course webpage.

Suppose we draw three cards from a deck and roll two dice. Answer the following questions.

What is the experiment?

The experiment is “drawing three cards from a deck and rolling two dice”.

What is “getting two-sixes and three-kings or five-one (in any order) one queen one king one ace”? Pick one (Event / Outcome / Sample Space)

Event.

Give an example of two mutually exclusive events.

Event A: Queen of Hearts / Queen of Spades / Queen of Diamonds / 6 / 5 Event B: Ace of Clubs / King of Clubs / Queen of Clubs / 4 / 4

What is the probability of getting four-three (in any order) in dice roll and three queens in card draw?

`#First roll can either be 3 or 4 and second roll should be the other # 2/6 * 1/6 #Getting the first Queen has probability of 4/52 #Getting the second Queen has probability of 3/51 #Getting the third Queen has probability of 2/50 2/6*1/6*4/52*3/51*2/50`

`## [1] 1.00553e-05`

How many different outcomes can there be? This time assume ordering is important (e.g. 6-1 and 1-6 are different outcomes).

`#Six outcomes per die #52 outcomes for the first card draw #51 outcomes for the second card draw #50 outcomes for the second card draw #Multiplication rule #You can also use permutation rule for cards 6*6*52*51*50`

`## [1] 4773600`

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

Any order.

`the_phrase<-"HOUSEPARTY" #No repetitive letters #Permutation rule factorial(nchar(the_phrase))`

`## [1] 3628800`

Vowels together?

`#4 vowels, 6 consonants #Assume all vowels are a single "letter". So 8 characters. #But vowels permutate within the single "letter". #Multiplication rule factorial(6+1)*factorial(4)`

`## [1] 120960`

Vowels in alphabetical order?

`#We start with all the permutations 10! #For any permutation there can be only one ordering of vowels. #For instance HOUSEPARTY is not valid but HAESOPURTY is valid #So remove invalid permutations with division factorial(10)/factorial(4)`

`## [1] 151200`

There should be no consecutive vowels?

`#There are 6 consonants, 4 vowels. # Assume Xs are consonants and .s are potent vowel places. # .X.X.X.X.X.X. #Consonants can permutate in any order so 6! there #7 places for vowels but only 4 vowels. # So it is a permutation of 4 out of 8 places. factorial(6)*(factorial(7)/factorial(7-4))`

`## [1] 604800`

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

```
# 11 characters.
# 6 vowels, 5 consonants
# 3 As, 2 Es, 2 Rs
```

Any order.

`#By the formula of permutation with repetitive letters #Assign the value to all_perms object all_perms<-factorial(11)/(factorial(3)*factorial(2)*factorial(2)) all_perms`

`## [1] 1663200`

Vowels together?

`#Assume all vowels are single "character" again. So 6 characters (factorial(5+1)/(factorial(2)))*(factorial(6)/(factorial(3)*factorial(2)))`

`## [1] 21600`

Vowels in alphabetical order?

`#Same as the last question. But be careful about identical vowels. all_perms/(factorial(6)/(factorial(3)*factorial(2)))`

`## [1] 27720`

There should be no consecutive vowels?

`#Same as the last question. But be careful about identical vowels. (factorial(5)/factorial(2))*(factorial(6)/factorial(6-6))/(factorial(3)*factorial(2))`

`## [1] 3600`

Suppose you are putting the top 12 basketball teams in 4 groups evenly (each group should consist of 3 teams). In how many different ways can you arrange the teams?

```
#It is either a chain of combinations or just grouping combination
choose(12,3)*choose(9,3)*choose(6,3)
```

`## [1] 369600`

There are 18 people; 10 from Izmir, 8 from Mugla.

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

`#Calculate as if no rules. It is the combination of 18 to 5. #Then remove the combinations of all Izmir or all Mugla people choose(18,5) - choose(10,5) - choose(8,5)`

`## [1] 8260`

In how many ways can you form a group of 3 people from Izmir and 4 people from Mugla?

`#Simply separate combinations with multiplication rule. choose(10,3)*choose(8,4)`

`## [1] 8400`