Today in Pre-Calculus

1
Today in Pre-Calculus

• Review Chapter 9 need a calculator
• Homework
• Go over Chapter 8 worksheet questions

2
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.

3
ExampleS

• There are ten drivers in a race. How many
  outcomes of first, second, and third place are
  possible?
• 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?

4
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?

5
Conditional Probability

• Notation P(AB) probability of A given B
• P(chocolate chipjar A)
• P(chocolate chipjar B)
• P(AB)
• P(jar Achocolate chip)

6
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
• nCn-rprqn-r
• Ex We roll a fair die four times. What is the
  probability that we roll
• All 3s
• b) no 3s
• c) Exactly two 3s

7
Binomial Theorem

• (a b)n nC0an nC1an-1b nC2an-2b2
• nCn-2a2bn-2 nCn-1abn-1 nCnbn
• Example (2x2 3y)4
• Find the x6y5 term in the expansion of (x 3y)11

8
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
• General formula an a1r(n1)

9
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,

10
Series

• Series the sum of the terms of a sequence a1,
  a2, ,an
• Written as
• Read as the sum of ak from k 1 to n.
• k is the index of summation

11
formulas
12
Example

• Write the sum of the following series using
  summation notation
• Example 1 13 17 21 49
• Example 2 1 8 27 (n1)3
• Example 3 3, 6, 12, , 12,288

13
Infinite Series

• An infinite series can either
• Converge if, as n increases, the series sum
  approaches a value (S)
• Diverge if as n increases, the series sum does
  NOT approach a value.

14
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,