Today

•

Today:

• – –

Reading:

Read Chapter 1 by next Tuesday Suggested problems (not to be handed in): 1.1, 1.2, 1.8, 1.10, 1.16, 1.20, 1.24, 1.28

Assignment 1

1. There are 38 equally likely positions on a roulette wheel numbered 0, 00, 1 ,2 …, 36. The 0 and 00 are green, the odd numbers are red and the even numbers are black. Find the probability that the outcome on a single spin is: a. red b. not black c. in the third dozen (i.e., 25, …, 36) 2. In a population of 100 undergraduate students, 20 are seniors, 25 are juniors and 30 are freshmen. Suppose that 6 seniors, 5 sophomores and three freshmen are smokers and that 7/9 upperclassmen (juniors and seniors) do not smoke. Find the probability that a randomly selected student is: a. a smoker b. a sophomore c. a freshman and a non-smoker d. not a senior and a non-smoker 3. Two tickets are drawn, without replacement, from a box that contains five tickets. The tickets are labeled 1,2,3,4 and 5. a. What is the sample space for this experiment? b. What is the probability of getting two odd tickets? c. What is the probability that the largest number drawn is equal to 4? 4. How many distinguishable arrangements can be made using the letters in the word MISSISSIPPI? 5. A committee of size three is to be selected, at random, from a group of four Asians Americans, four African Americans, and twelve European Americans. What is the probability that every group is represented on the committee?

Combinatorics (Section 1.4)

• In the equally likely case, computing probabilities involves counting the number of outcomes in an event • This can be hard…really • Combinatorics is a branch of mathematics which develops efficient counting methods

Combinatorics

• • Consider the rhyme

As I was going to St. Ives I met a man with seven wives Every wife had seven sacks Every sack had seven cats Every cat had seven kits Kits, cats, sacks and wives How many were going to St. Ives?

Answer:

Example

• In three tosses of a coin, how many outcomes are there?

Multiplication Principle

• • Let an experiment E E 1 ,E 2 , …, E k , where E be comprised of smaller experiments i has

n i

outcomes The number of outcome sequences in E is (

n 1 n 2 n 3

…

n k

) • Example (St. Ives re-visited)

Example

• In three tosses of a coin, how many outcomes are there?

•

Tree Diagram

Example

• In a certain state, automobile license plates list three letters (A-Z) followed by four digits (0-9) • How many possible license plates are there?

Example

• Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit) • Suppose you are going to draw 5 cards, one at a time, with replacement (with replacement means you look at the card and put it back in the deck) • How many sequences can we observe

Permutations

• In previous examples, the sample space for E i does not depend on the outcome from the previous step or sub-experiment • • The multiplication principle applies only if the

number

of outcomes for E i is the same for each outcome of E i-1 That is, the outcomes might change change depending on the previous step, but the number of outcomes remains the same

Permutations

• When selecting object, one at a time, from a group of

N

objects, the number of possible sequences is: • The is called the number of permutations of

N

things taken

n

at a time • • Sometimes denoted N P n Can be viewed as number of ways to select where the order matters

N

things taken

n

at a time

Example

• Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit) • Suppose you are going to draw 5 cards, one at a time, without replacement • How many permutations can we observe

Counting Patterns

• Consider the word

minimum

• How many permutations of the letters are there?

• How many distinguishable ways are there to to arrange these letters?

Counting Patterns

• The number of distinct sequences of

N

type 1,

m 2

are are of type 2, …,

m k

objects where are are of type k is:

m 1

are are of  

m

1 ,

N m

2 ,...

m k

  

N

!

m

1 !



m

2 !

 ...



m k

!

• Note:

N= m 1 + m 2 + …+ m k

Counting Patterns

• Consider the word

minimum

• How many permutations of the letters are there?

• How many distinguishable ways are there to to arrange these letters?

Example

• In a certain state, automobile license plates list three letters (A-Z) followed by four digits (0-9) • How many possible license plates are there with 7 distinct characters?

Combinations

• If one is not concerned with the order in which things occur, then a set of

n

objects from a set with

N

objects is called a combination Example – Suppose have 6 people,3 of whom are to be selected at random for a committee – – The order in which they are selected is not important How many distinct committees are there?

Combinations

• The number of distinct combinations of

n

objects selected from

N

objects is:  

N n

   (

N



N

!

n

)!

n

!



N C n

• • • • “N choose n” Note:

N!=N(N-1)(N-2)…1

Note:

0!=1

Can be viewed as number of ways to select

N

things taken

n

at a time where the order

does not

matter

Combinations

Example – Suppose have 6 people, 3 of whom are to be selected at random for a committee – – The order in which they are selected is not important How many distinct committees are there?

Example

• A committee of size three is to be selected from a group of 4 Democrats, 3 Independents and 2 Republicans • How many committees have a member from each group?

• What is the probability that there is a member from each group on the committee?