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Counting (Combinatorics)

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Equally Likely Outcomes• Theoretical (equally likely)

• ExampleA song is chosen at random from a person’s mp3 player. If there are 10 alternative, 5 blues, 7 jazz and 25 rock songs on the player, what is the probability that a rock song is played?

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Terminology

• Ordered vs. unordered– Distinguishable

• With replacement vs. without replacement– replication

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Counting Techniques  ordered unordered

With replacement

Without replacement

 

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Basic Counting Rule (BCR)

Requirements: ordered, with replacementExample: How many pairs are there when you

roll a 4-sided die and a 3-sided die?Calculation: If the first element of an ordered

k-tuple can be selected in n1 ways, the second element in n2 ways up to the kth element in nk ways, then there are possible k-tuples.

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Product Rule: Examples

1) How many pairs are there when you roll a 4-sided die and a 3-sided die?

2) How many different possible types of students are there in a STAT 350 class if the students can be classified by class (Freshmen, Sophomores, Juniors or Seniors) and by major (Math, CS, Statistics, Actuarial Science, Engineering, Physical Science, Other)

3) How many different possibilities are there when you roll a 4-sided die, a 3-sided die and a 10-sided die?

4) Consider the random experiment of rolling one 4-sided die. If the die is rolled 2 times, how many possible outcomes are there? How many possible outcomes are there if you would roll the die n times?

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Counting Techniques  ordered unordered

With replacement

Product Rule(multiplication, tree diagrams)

Without replacement

 

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Permutation

Requirements: ordered, without replacementThe number of permutations of size k that be

formed from a set of n distinct objects is denoted by Pk,n.

Example: How many different ways can we draw 4 cards from the 13 spades in the deck of cards without replacement?

Calculation:

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Permutation: Examples1) How many different ways can we draw 4 cards from the

13 spades in the deck of cards without replacement?2) How many possible ways can you draw an ordered

sample of size 0 from a population of size n?3) How many possible ways can you draw an ordered

sample of size n from a population of size n?4) In a horse race consisting of 10 horses, how many

different ways are there to choose the horses that finish first, second and third?

5) If there are 20 students in a club, how many unique ways can the president, vice president, secretary and treasurer be elected (assuming that no person can hold more than one office)?

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Counting Techniques  ordered unordered

With replacement

Product Rule(multiplication, tree diagrams)

Without replacement

Permutation

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Combination

Requirements: unordered, without replacementThe number of combinations of size k that be

formed from a set of n distinct objects is denoted by Ck,n, or read n choose k.

Example: How many ways can I choose 3 of the 6 different type of dark red dice?

Calculation:

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Combination

1) How many ways can I choose 3 of the 6 different type of dark red dice?

2) How many possible ways can you draw an unordered sample of size 0 from a population of size n?

3) How many possible ways can you draw an unordered sample of size 1 from a population of size n?

4) In a horse race consisting of 10 horses, how many different ways are there to choose the horses that finish in the money?

5) If there are 20 students in a club, how many unique ways can you choose for four students to go to a conference?

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Counting Techniques  ordered unordered

With replacement

Product Rule(multiplication, tree diagrams)

Without replacement

Permutation Combination

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Probability - Counting

I have 10 dice in the bag, 6 dark red, 1 red and 3 white. What is the probability that the 3 dice are:

a) 1 white and 2 dark red?b) all red?

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Counting - Procedure

1. Construct the sample space where each outcome is equally likely to happen.

2. Determine the event of interest.3. Count the number of outcomes of the event

and the total number of outcomes.4. Find the probability by:

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Complicated Situations• Poker – 5-card Draw

– Full house– 2-pair

• Mixture of ordered and unorderedAt a track meet, three North American countries are represented: the United States, Canada and Mexico. Eight athletes are competing in the 100-yard dash, three from the United states, two from Canada, and three from Mexico. If you only consider what country an athlete comes from, how many different ways can athletes finish?

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Examples for Counting Techniques1. Roll two 8-sided dice. What is the probability that the

sum of the two numbers is 5? (0.0625)2. Draw two cards from a suit of 13 cards (say diamonds),

what is the probability that the sum of the two cards is even? (A = 1, J = 11, Q = 12, K = 13)? (0.462)

3. The IRS decides that it will audit the returns of 3 people from a group of 18. If 8 of the people are women, what is the probability that all 3 of people audited are women? (0.0686)

4. Arizona places consist of three digits followed by three letters. What is the probability that a particular license plate doesn’t have any repeating digits or letters? (0.639)

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What method to use?1.How many batting orders are there for 9 players on

a baseball team? [362,880]2.In a math club at Purdue with 20 members, 3

people can go to a national conference. How many different ways can these people be chosen? [1140]

3.Out of 10 people in the U.S., 2 live in the Northeast, 3 live in the Midwest, 3 live in the South, and 2 live in the West. How many different ways can these 10 people be assigned to these regions? [25,200]

X

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What method to use? (cont)5. At a movie festival, a team of judges is to pick the

first, second, and third place winners from the 18 files entered. How many possible ways are there to choose the winners? [4896]

6. If a password contains 7 lower case letters, how many possible passwords are there? [8,031,810,176]

7. In 5-card draw, how many different hands can you receive? In 5-card draw, the order of the cards is not important. [2,598,960]

X

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What method to use? (cont)9. For a certain airport, there are three runways that are used, A, B

and C. Out of the next 15 jets, how many ways can 10 land on runway A, 2 land on runway B and the rest land on runway C? [30,030]

10. A multiple choice exam has 40 questions each of which have 5 possible choices. How many possible combinations of answers are there? [9.09 X 1027]

11. You are groups of 5 students each from your class of 20 students. If each of these groups are performing different assignments, how many ways can you arrange your class? [1.17 X 1010]

12. A statistics professor wants to do perform in depth interviews to see how she is teaching. So she has decided to choose 5 students from her class of 40. How many different possibilities are there? [658,008] X

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What method to use? (cont)13. The menu at a restaurant has five choices of a

beverage, three different salads, six entrées and four deserts. How many different meals are possible? [360]

14. The sales manager of a clothing company needs to assign seven salespeople to seven different territories. How many possibilities are there for the assignments? [5040]

X

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What method to use? (cont)17. You roll 6 dice where the order is important,

the first two are 4-sided, the next two are 6-sided and the last two are 10 sided.

a) How many different combinations are possible? [57,600]

b) What is the total number of combinations if no number can occur twice on the same size die? [32,400]

c) What is the total number of combinations if no number can occur twice on any size die? [4320] X