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.
ExampleS1)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) =
( )
( )
P A and B
P B
Binomial DistributionLet 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:a)All 3’s
b) no 3’s
c) Exactly two 3’s
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
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(n–1)
1 1 1 1, , , ,....
2 4 8 16
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}
• Written as:
• Read as “the sum of ak from k = 1 to n”.
• k is the index of summation5
1
2k
k
1
n
kk
a
6
3
2k
k
formulas
1
1
1:
1
nn
kk
a rGeometric a
r
1 11
: 2 ( 1)2 2
n
k nk
n narithmetic a a a a n d
100
1
:k
Example k
110
1
1: 28
2
k
k
Example
ExampleWrite 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 SeriesAn 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.
1
1
3? , . 7
5
k
k
Does this series converge If so give the sum
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,