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