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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( A and B) P( 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 rqn-r C p n 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 = nC0an + nC1an-1b + nC2an-2b2 + … + 2bn-2 + C abn-1 + C bn C a n n-2 n n-1 n n Example: (2x2 – 3y)4 = Find the x6y5 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: an = a1 + (n-1)d • Geometric: a sequence in which there is a common ratio (or quotient) between every pair of successive terms. Example: 1 , 1 , 1 , 1 ,.... 2 4 8 16 General formula: an = a1r(n–1) Explicitly Defined Sequence • A formula is given for any term in the sequence Example: ak = 2k - 5 Find the 20th 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 {a1, a2, …,an} n Written as: ak k 1 Read as “the sum of ak from k = 1 to n”. k is the index of summation 5 2k k 1 6 k 2 k 3 formulas n n n arithmetic : ak a1 an 2a1 (n 1)d 2 2 k 1 100 Example : k k 1 n Geometric : ak k 1 a1 1 r n 1 r 1 Example : 28 2 k 1 10 k 1 Example Write the sum of the following series using summation notation: Example 1: 13 + 17 + 21 + … + 49 Example 2: 1 + 8 + 27 + … + (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. ¥ k-1 a ir converges iff r < 1 å k=1 ¥ If it does converge, the å a ir k=1 k-1 a1 = 1- r 3 Does this series converge ? If so, give the sum. 7 5 k 1 k 1 Homework • • • • Pg 708: 2,4,11,19,21,22 Pg 715: 13,15,19,21 Pg 728: 31,33, 45-50 Pg 787: 55-58, 63,70,77-81odd,83,84,