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Probability

Probability. Factorial, Permutations, Combinations Week 6 TEST # 2 – Next week!. Permutations. are arrangements of n (number of objects) in a specific order. With permutations “order matters”!

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Probability

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  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! = 5*4*3*2*1 = 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! = 3*2*1 = 6 ways.

  5. 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?

  7. What about: MISSISSIPPI?

  8. 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 Prn  is the total number of objects  ris the number of objects chosen(want)

  9. n Pr Formula:n Pr = n!/(n-r)! (ex 1) 6 P4 = 6!/(6-4)! = 6!/2! = (6*5*4*3*2!)/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 =   5·4·3  =  60          

  10. 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

  11. Statistics Prize Money! • It’s 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?

  12. 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?

  13. 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.

  14. 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 ofDeb, Lydia, and Jessicais THE SAME as Jessica, Lydia, and Deb. Thus, there is only one group.

  15. 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 combo’s 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. Permutation       versus       Combination 1. Picking a team captain, pitcher, and shortstop from a group. 1. Picking three team members from a group. 2.  Picking your favorite two colors, in order, from a color brochure. 2.  Picking two colors from a color brochure. 3.  Picking first, second and third place winners. 3.  Picking three winners. What’s the diff’?

  17. 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?

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