Probability

1
Probability

• Factorial, Permutations, Combinations
• Week 6
• TEST 2 Next week!

2
Permutations

• are arrangements of n (number of objects) in a
  specific order.
• With permutations order matters!
• Problems involve the words order, different,
  arrange, specific, place, position, rank,
  anything that has to do with a specific spot.

3
Factorial Notation

• Is a shorthand way to express multiplication of
  decreasing, consecutive integers.
• Formula n! n(n-1)(n-2)..(1)
• (Ex. 1) 5! 54321 120.
• (Ex. 2) 0! 1
• (Ex. 3) 9!

4
Uses for Factorial

• How many different ways can I arrange the letters
  in my first name JOE ?
• JOE, JEO, OJE, OEJ, EJO, EOJ 6ways
• There are three letters, thus, 3!
• 3! 321 6 ways.

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More Examples

• (Ex. 1) How many ways can a coach arrange a
  line-up of 6 baseball players?
• (Ex. 2) In how many different ways can I
  re-arrange the seating of 8 people?
• (Ex. 3) How many different ways can I arrange 10
  questions on a quiz?

6
More Examples

• (Ex. 4) How many different ways can I arrange the
  letters in the word MATH?
• (Ex. 5) How many different ways can I arrange the
  letters in the word PASS?
• (Ex. 6) How many different ways can I arrange the
  letters in the word TEXTBOOK?
• (Ex. 7) How many different ways can I arrange the
  letters in the word STATISTICS?

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What about

• MISSISSIPPI?

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Smaller Arrangements of Larger Group

• Permutation Rule is the arrangement of n
  objects in a specific order, using only r at a
  time.
• The notation for a permutation n Pr    n  is
  the total number of objects    r   is the number
  of objects chosen (want)

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n Pr

• Formula n Pr n!/(n-r)!
• (ex 1) 6 P4 6!/(6-4)! 6!/2! (65432!)/2!
  
• 360
• (ex 2) 8 P3
• (ex 3) 5 P5
• (ex 4) How many different 3-digit numerals can be
  made from the digits  4, 5, 6, 7, 8   if a digit
  can appear just once in a numeral?        5 P3 
     543    60          

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Sabres Line-up

• In how many ways can Lindy Ruff arrange 13
  forwards in front lines of 3 players?
• (Order matters here because there is a center, a
  right wing, and a left wing.)
• 13 P3 1,716 ways

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Statistics Prize Money!

• Its time for the big pay-out! The college is
  going to pay-out three places to students in
  Thursday night Statistics.
• 1st 5,000 2nd 3,000 3rd 1,000
• With 26 students in the class, find the
  following
• a) P(1st Place)
• b) P(Winning)
• c) How many different arrangements of winners can
  be made from our class?

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Special Arrangements

• How many different license plates can NY State
  issue if they are to have 3 letters followed by 4
  numbers?
• How many different license plates can NY State
  issue if they are to have 3 different letters
  followed by 4 different numbers?

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Special Arrangements

• A new area code is being created. How many phone
  numbers are being created if the following
  specifications are met?
• The 1st number cannot be a ZERO or a ONE
• The first three cannot be 911 or 411.

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Combinations

• Combination  A set of objects in which position
  (or order) is NOT important.
• How many different groups of 3 can be formed
  including Deb, Lydia, and Jessica?
• (3 people)
• In a combination, the trio of Deb, Lydia, and
  Jessica is THE SAME as Jessica, Lydia, and Deb.
  Thus, there is only one group.

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n Cr Formula n Cr n!/(r!(n-r)!) (ex 1) 6 C4
15 (ex 2) 9 C3 (ex 3) 5 C5 (ex 4) How many
different 3-letter combos can be made from the
letters  A, B, C, D, E   if a letter can appear
just once in a combo?        5 C3    
10          
16
Whats the diff?
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Prize Winners

• (Ex 1) A raffle has 20 entries. The prizes
  include 5 gift certificates, all for 20 each.
  How many different groups can be selected to
  claim the prizes?
• (Ex 2) 15 people placed their names in a hat to
  win trip to beautiful downtown Sanborn. If the
  prize commission is only choosing 8 winners, how
  many groups can be formed?

18
Sabres Line-up

• In how many groups can Lindy Ruff arrange 13
  forwards in front lines of 3 players?
• (If order does not matter, each player could
  change up to be a center, a right wing, and a
  left wing.)
• 13 C3

19
Form a committee

• (Ex 1) A committee is to be formed consisting
  of 3 people. There are 5 people to choose from,
  how many different committees can be created?
• (Ex 2) A committee is to be formed consisting
  of 5 people. There are 12 people to choose from,
  how many different committees can be created?

20
Special Emergency Committee

• A new committee is to be formed from a group of
  20 college students. It is to have 6 members and
  it must contain an equal amount of boys as girls.
  There are 12 boys in the original group. How
  many groups can be formed?