Today in Pre-Calculus Review Chapter 9 – need a calculator Homework Go over Chapter 8 worksheet questions

Combinatorics An arrangement of objects in a specific order or selecting all of the objects. An arrangement of objects in which order does not matter. Difference between permutations and combinations: –Combinations: grouping of objects –Permutation: putting objects in specific places or positions, or selecting all of the objects.

ExampleS 1)There are ten drivers in a race. How many outcomes of first, second, and third place are possible? 2)In a study hall of 20 students, the teacher can send only 6 to the library. How many ways can the teacher send 6 students?

Conditional Probability & Tree Diagrams Two identical cookie jars are on a counter. Jar A contains 2 chocolate chip and 2 peanut butter cookies, while jar B contains 1 chocolate chip cookie. Selecting a cookie at random, what is the probability that it is a chocolate chip cookie?

Conditional Probability Notation: P(A|B) probability of A given B P(chocolate chip|jar A)= P(chocolate chip|jar B)= P(A|B)= P(jar A|chocolate chip) =

Binomial Distribution Let p be the probability of event A and q be the probability of event A not occurring given n trials. Then the probability A occurs r times is n C n-r p r q n-r Ex: We roll a fair die four times. What is the probability that we roll: a)All 3’s b) no 3’s c) Exactly two 3’s

Binomial Theorem (a + b) n = n C 0 a n + n C 1 a n-1 b + n C 2 a n-2 b 2 + … + n C n-2 a 2 b n-2 + n C n-1 ab n-1 + n C n b n Example: (2x 2 – 3y) 4 = Find the x 6 y 5 term in the expansion of (x + 3y) 11

Sequences Arithmetic Sequence: a sequence in which there is a common difference between every pair of successive terms. Example: 5,8,11,14 General formula: a n = a 1 + (n-1)d Geometric: a sequence in which there is a common ratio (or quotient) between every pair of successive terms. Example: General formula: a n = a 1 r (n–1)

Explicitly Defined Sequence A formula is given for any term in the sequence Example: a k = 2k - 5 Find the 20 th term for the sequence Write the explicit rule for the sequence 55, 49, 43, … Write the explicit rule for the sequence 5, 10, 20, …

Series Series: the sum of the terms of a sequence {a 1, a 2, …,a n } Written as: Read as “the sum of a k from k = 1 to n”. k is the index of summation

formulas

Example Write the sum of the following series using summation notation: Example 1: … + 49 Example 2: … + (n+1) 3 Example 3:3, 6, 12, …, 12,288

Infinite Series An infinite series can either: 1)Converge – if, as n increases, the series sum approaches a value (S) 2)Diverge – if as n increases, the series sum does NOT approach a value.

Homework Pg 708: 2,4,11,19,21,22 Pg 715: 13,15,19,21 Pg 728: 31,33, Pg 787: 55-58, 63,70,77-81odd,83,84,