(Lesson 13: Counting; 4-6) 4.25

LESSON 13: COUNTING (SECTION 4-6)

PART A: FUNDAMENTAL COUNTING RULE

Let’s say there are two decisions to be made: Decision A and Decision B.If there are m possible choices for Decision A

A

1, A

2,…, A

m( ) and

n possible choices for Decision B

B1, B

2,…, B

n( ) , regardless of how A is decided,

then there are mn possible ways to decide both A and B.

Envision a possibility tree or a grid:

A \B B1 B2

… Bn

A1

A2

Am

This multiplication rule extends to more than two decisions.

(Lesson 13: Counting; 4-6) 4.26

Example 1

Assume that a password must have the following characteristics:

• It must be (exactly) four characters long.

• It must begin with an uppercase letter.

• Each of the other characters can be an uppercase letter or a digitfrom 0 to 9.

How many possible passwords are there?

Solution to Example 1

There are 26 possibilities (the uppercase letters) for the first character.

There are 36 possibilities (the uppercase letters and the digits 0-9) for eachof the others.

The number of possible passwords is:

26 × 36 × 36 × 36 = 1,213,056

1st 2nd 3rd 4th

Note: If you write 26 ⋅363 , remember that exponentiation precedesmultiplication in the order of operations.

Example 2 (Follow-Up)

If a password is randomly selected, what is the probability that it is“IH8U”?

Solution to Example 2

The 1,213,056 possible passwords are equally likely to be selected, so the

probability of getting a particular one (such as “IH8U”) is

1

1,213,056.

(Lesson 13: Counting; 4-6) 4.27

PART B: PERMUTATIONS (ORDERINGS)

(Lesson 13: Counting; 4-6) 4.28

(Lesson 13: Counting; 4-6) 4.29

PART C: PARTIAL PERMUTATIONS

Look for n Pr on calculators.

The formula is most useful when r is large.

(Lesson 13: Counting; 4-6) 4.30

PART D: COMBINATIONS

Look for nCr on calculators.

(Lesson 13: Counting; 4-6) 4.31

Method 2 (Helps with Ch.5.)

Before the race, randomly assign the numbers 1 through 7 to therunners. After the race, three win (W) and four lose (L).

Related Question: Let’s say you are given a list of seven essay questions,and you must choose three of them to write on. How many ways are there tochoose the three questions? The answer, again, is 35.

Related Question: How many ways are there to get exactly three questionsright on a test of seven questions? The answer, again, is 35.

(Lesson 13: Counting; 4-6) 4.32

PART E: HOW DO I DECIDE WHICH RULE TO USE?

Two key questions:

1) Are possibilities being reduced for future tasks/decisions? If not, maybe just usethe Fundamental Counting Rule (Part A). If we have the same numbers ofpossibilities (choices) for different decisions, then a power may be involved.

2) If possibilities are being reduced, does order matter among the winners/chosenitems, or do we distinguish among them? If so, consider using permutations(Part C), with factorials corresponding to the special case of “complete”permutations (Part B). If order does not matter, consider using combinations(Part D).

(Lesson 13: Counting; 4-6) 4.33

The prior probability is low; remember our second example inSection 4-5.

(Lesson 14: Random Variables; 5-2) 5.01

CHAPTER 5:PROBABILITY DISTRIBUTIONS

LESSON 14: RANDOM VARIABLES (SECTION 5-2)

PART A: “WHO WANTS TO BE A MILLIONAIRE?” EXAMPLE

(Lesson 14: Random Variables; 5-2) 5.02

(Lesson 14: Random Variables; 5-2) 5.03

PART C: VAR(X) AND SD(X)

We will simplify the approach in Triola.

(Lesson 14: Random Variables; 5-2) 5.04

(Lesson 15: Binomial Distributions; 5-3) 5.05

LESSON 15: BINOMIAL DISTRIBUTIONS (SECTION 5-3)

PART A: WHAT IS A BINOMIAL EXPERIMENT?

Similar examples:

• 1/6 of all Americans have cooties. Five Americans are randomly sampled.Let X = the number in the sample with cooties.

• A multiple-choice test has five questions, each with six options (A-F).A student guesses randomly on all five questions.Let X = the number of correct answers.

(Lesson 15: Binomial Distributions; 5-3) 5.06

Note:

P 5( ) ≈ 0.000129

(Lesson 15: Binomial Distributions; 5-3) 5.07

PART B: HOW DO WE FIND THE PROBABILITIES?

1) Software / TI calculators

2) Table A-1

(pp.609-611 in the 3rd edition of Triola)

Provided on exams.Use it when you can.Up to n = 15 , some values of p.

3) Binomial probability formula

(Lesson 15: Binomial Distributions; 5-3) 5.08

(Lesson 15: Binomial Distributions; 5-3) 5.09

(Lesson 15: Binomial Distributions; 5-3) 5.10

Note: P 5( ) ≈ 0.000129

(Lesson 15: Binomial Distributions; 5-3) 5.11

PART C: PASCAL’S TRIANGLE

(Lesson 15: Binomial Distributions; 5-3) 5.12

PART D: SAMPLING RULE

(See Lesson 11.)

PART E: USING TABLE A-1

(pp.609-611 in the 3rd edition of Triola)

Note: “At least 4” can be rephrased as “4 or more” or “no fewer than 4.”

(Lesson 15: Binomial Distributions; 5-3) 5.13

Think About It: Why are the probabilities skewed towards the highend?

We want:

(Lesson 15: Binomial Distributions; 5-3) 5.14

(Lesson 15: Binomial Distributions; 5-3) 5.15

PART F: WHEN IS AN EVENT “UNUSUAL”?

Rare Event Rule for Inferential Stats (modified from Triola)

(Lesson 15: Binomial Distributions; 5-3) 5.16

Note: P 49( ) ≈ 7.80%

(Lesson 15: Binomial Distributions; 5-3) 5.17

Analysis #2 (one-tailed) (use in HW)

Analysis #3 (two-tailed; we use symmetry)

Analysis #4 (z scores)

See the next Lesson!

(Lesson 16: Mean, VAR, and SD for Binomial Distributions; 5-4) 5.18

LESSON 16: MEAN, VAR, AND SDFOR BINOMIAL DISTRIBUTIONS (SECTION 5-4)

(Lesson 16: Mean, VAR, and SD for Binomial Distributions; 5-4) 5.19

(Lesson 16: Mean, VAR, and SD for Binomial Distributions; 5-4) 5.20

Coin Example (Analysis #4; see previous Lesson)

(Chapter 5: Extra) 5.21

Extra: Why Does the Binomial Probability Formula Work?