Chapter 12 PROBABILITY Counting, Combinations, Permutations, Probability, Binomial Theorem

Fundamental Counting Principle Two events: one occurs m ways and the other occurs n ways… They occur together mn ways EXAMPLE: How many different 7 digit phone numbers are possible if the first digit cannot be 0 or 1?

PERMUTATIONS An ORDERING of n objects: n! Permutations of n objects taken r at a time Permutations with repetition

Example Questions 1.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 Questions 2.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 Questions 3.Find the number of distinguishable permutations of the letters in POPPY

Example Questions 4.Find the number of distinguishable permutations of the letters in WATERFALL

COMBINATIONS Selection of objects – NO ordering EXAMPLE: How many 5-card hands are possible when using a standard deck?

BINOMIAL 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 Triangle Write Pascal’s Triangle to the ( ) 7 line:

Example Questions 5.Expand (x - 5) 3

Example MC Questions 6.What is the coefficient of x 5 in the expansion of (2x - 3) 6 ? A) 6B) -36C) 192 D) -576 E) -1152

PROBABILITY 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 Questions 7. 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) 4B) 12C) 24 D) 120E) 360

Example MC Questions 8.In how many ways can a 5 person committee be chosen from a group of 8 people? A) 15B) 40 C) 56 D) 336 E) 6720

Example MC Questions 9.Events A and B are independent, P(A)=0.9 and P(B)=0.4. What is P(A and B)? A) 0.4B) 0.9 C) 1.3D) 0.36 E) 3.6

Example MC Questions 10.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).2B) 0.25C) 0.28 D) 0.4E) 0.14 HamburgerHot DogsChicken Sandwich Mustard40220 Ketchup10 5

Example MC Questions 11.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) 1/3B) 1/2C) 5/16 D) 5/32E) 5/64