chapter 12 probability counting, combinations, permutations, probability, binomial theorem

Chapter 12 Chapter 12 PROBABILITYPROBABILITY

Counting, Combinations, Counting, Combinations, Permutations, Probability, Permutations, Probability,

Binomial TheoremBinomial Theorem

Fundamental Fundamental Counting PrincipleCounting Principle

Two events: one occurs Two events: one occurs mm ways ways and the other occurs and the other occurs nn ways… ways…They occur together They occur together mnmn ways ways

EXAMPLE: How many different 7 digit phone numbers are possible if the first digit cannot be 0 or 1?

PERMUTATIONSPERMUTATIONS• An ORDERING of n objects: n!• Permutations of n objects taken r at

a time

• Permutations with repetition

Example QuestionsExample Questions1. There are 8 movies you would like

to see currently showing in theaters. In how many different ways can you see all 8 of the movies?

Example QuestionsExample Questions2. There are 8 movies you would like

to see currently showing in theaters. In how many ways can you choose a movie to see this Saturday and one to see this Sunday?

Example QuestionsExample Questions3. Find the number of distinguishable

permutations of the letters in

POPPY

Example QuestionsExample Questions4. Find the number of distinguishable

permutations of the letters in

WATERFALL

COMBINATIONSCOMBINATIONS• Selection of objects – NO ordering

• EXAMPLE: How many 5-card hands are possible when using a standard deck?

BINOMIAL THEOREMBINOMIAL THEOREM• Pascal’s Triangle – coefficients of the

binomial expansion• Sum of the exponents in each term

must add to the original power

EXAMPLE: Expand (x-5)4

Pascal’s TrianglePascal’s TriangleWrite Pascal’s Triangle to the ( )7 line:

Example QuestionsExample Questions5. Expand (x - 5)3

Example MC QuestionsExample MC Questions6. What is the coefficient of x5 in the

expansion of (2x - 3)6?

A) 6 B) -36 C) 192 D) -576 E) -1152

PROBABILITYPROBABILITY• Probability:

• P(A or B) = P(A) + P(B) – P(A and B)P(A and B) = P(A) • P(B)Complement: P(A’) = 1 – P(A)

Example MC QuestionsExample MC Questions7. You have 6 best friends, but only 4

cupcakes. In how many different orders can you give four of your six best friends cupcakes?

A) 4 B) 12 C) 24D) 120 E) 360

Example MC QuestionsExample MC Questions8. In how many ways can a 5 person

committee be chosen from a group of 8 people?

A) 15 B) 40C) 56 D) 336E) 6720

Example MC QuestionsExample MC Questions9. Events A and B are independent,

P(A)=0.9 and P(B)=0.4. What is P(A and B)?

A) 0.4 B) 0.9C) 1.3 D) 0.36E) 3.6

Example MC QuestionsExample MC Questions10.The table below shows the number of

students that like mustard and ketchup with their food.

What is the probability that students who eat hamburgers choose ketchup as their condiment?A) .2 B) 0.25 C) 0.28D) 0.4 E) 0.14

Hamburger Hot Dogs Chicken Sandwich

Mustard 40 2 20

Ketchup 10 10 5

Example MC QuestionsExample MC Questions11.What is the probability that in a

family of 6 children exactly three are boys? Assume a boy and girl are equally likely to occur.

A) 1/3 B) 1/2 C) 5/16D) 5/32 E) 5/64