Transcript Document
a n = a 1 + (n – 1)d
S
a
1
1 r
a n = a 1 • r (n – 1)
Chapter 9 Review
n a k n a 1 a n 2 n 2 2a 1 n a k 1 n Pre-Calculus 1/31/2007
Pre-Calculus
Probability Review
(Sections 9.1 – 9.3)
1/31/2007
Fundamental Principle of Counting
(Multiplication Principle) n! = n(n – 1)(n – 2)(n – 3)… (2)(1) How many ways can you arrange:
A, B, C, D, E (5)(4)(3)(2)(1) = 120
Pre-Calculus 1/31/2007
Permutations
Order is important!!!
using “n” objects to fill “r” blanks in order.
permutations of “n” objects taken “r” at a time: n P r = n!
(n – r)!
How many ways can 6 runners finish 1 – 2 – 3?
6 P 3 = 6!
(6 – 3)!
= 720 6 = 120
Pre-Calculus 1/31/2007
Combinations
Order is NOT important!!!
only interested in the ways to select the “r” objects regardless of the order in which we arrange them.
n r
n C r = n!
(n – r)!
•
r!
How many ways can 2 cards be picked from a deck of 10: 10 C 2 = 10!
(8)!
•
2!
= 3,628,800 = 45 40320
•
2
Pre-Calculus 1/31/2007
Subsets of an “n” – set 2
n There are ________ subsets of a set with n objects (including the empty set and the entire set). Example: DiMaggio’s Pizzeria offers patrons any combination of up to 10 different pizza toppings. How many different pizzas can be ordered if we can choose any number of toppings (0 through 10)?
We could add up all the numbers of the form for r = 0, 1, …, 10 but there is an easier way. In considering each option of a topping, we Therefore the number of different possible pizzas is:
2
n
= 2
10
= 1024
Pre-Calculus 1/31/2007
Pre-Calculus
Binomial Theorem
n a n n
Don’t forget:
n n C r n a b r n b n 1/31/2007
Also, recall that in mathematics, the word or signifies addition; the word and signifies multiplication .
Find the probability of selecting an ace or a king from a draw of one card from a standard deck of cards.
4 52 4 52 8 52 2 13 or 0.154
Find the probability of selecting an ace and a king from a draw of one card from a standard deck of cards.
4 4 52 52 16 2704 1 169 or 0.0059
Pre-Calculus 1/31/2007
Venn Diagrams girls students sports 0.36 0.18 0.23
0.23
Pre-Calculus
sample space (all students) subsets to represent “girls” and “sports” “boys” “no sports” decimals 1
1 (54%) 3 0.18
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0.25
0.5
1 0.125 + 0.125 + 0.5 = 0.75
conditional probability dependent P(A B) of the event A, given that event B occurs 2/4 or 0.5
Pre-Calculus
the probability 1 along the branches that come out of the two jars
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P(A) • P(B A) the ends of the branches conditional probability formula
Pre-Calculus P(A) P( jar A and chocolate chip P(chocolate chip) ) 0.75
0.25
0.75
1 3 1/31/2007
binomial distribution binomial Theorem
5 6 1 6 1 6 1 6 1 6 1 6 1 6 4 0.00077
5 6 4 0.48225
1 6 5 6 2 0.01929
4 C 2 = 6
1 5 2 6 6
6
4 C 2 1 6 5 6 2 0.11574
Pre-Calculus 1/31/2007
Homework Answers
# 54, 56, 61, 65, 68, 70, 76, 77 , 17, 23 (p. 748 – 749)
5 (2x) 5 5 5 (2x y) 5 2 5 3 5 (2x)(y) 4 5 (y) 5 (1)32x 5 4 (5)16x y 3 (10)8x y 2 2 (10)4x y 3 (5)2xy 4 (1)y 5 32x 5 4 80x y 3 80x y 2 2 40x y 3 10xy 4 y 5 11 8 3 (x 2) 11 8 1320x 8 Pre-Calculus 1/31/2007
Pre-Calculus
Probability Review
Questions
1/31/2007
Probability Exercise
Suppose there is a 70% chance of rain tomorrow. If it rains, there is a 10% chance that all of the rides at an amusement park will be operating. If it doesn’t rain, there is a 95% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?
.355
Pre-Calculus 1/31/2007
Probability Exercise
Binomial Theorem: Bubba rolls a fair die 6 times. What is the probability that he will roll exactly two 2’s?
6 C 2 1 6 5 6 4 0.2009
Pre-Calculus 1/31/2007
Probability Exercise
There are 20 runners on a track team. How many groups of 4 can be selected to run the 4 x 100 relay?
4845 20 C 4
How many ways can 4 runners be selected to run 1 st – 2 nd – 3 rd – 4 th ?
20 P 4 116280 Pre-Calculus 1/31/2007
Probability Exercise
A fair coin is tossed 10 times.
Find the probability of tossing HHHHHTTTTT.
1 2 10 1 1024
Find the probability of tossing exactly 5 tails in those 10 tosses
10 C 5 1 2 10 252 1024 .2461
Pre-Calculus 1/31/2007
Pre-Calculus
Review Question
# 43 (p. 748)
1/31/2007
Pre-Calculus
Probability Exercise
Find the x 4 term in the expansion of:
(4x y) 5 1 5 1 1/31/2007
Probability Exercise
License plates are created using 3 letters of the alphabet for the first 3 characters and 4 numbers for the last 4 characters.
How many possible different license plates are there if the letters and numbers are NOT allowed to repeat?
78,624,000 Pre-Calculus 1/31/2007
Probability Exercise
A spinner, numbered 1 through 10, is spun twice.
What is the probability of spinning a 1 and a 10 in any order?
What is the probability of not spinning the same number twice?
2 1 10 10 1 50 Pre-Calculus 1/31/2007
Probability Exercise
Expand:
(3x 2y) 4 4 4 C (3x) (2y) 0 0 4 3 4 3 C (3x) (2y) 1 1 4 2 C (3x) (2y) 2 2 3 4 4 4 (3x) 4 3 2 2 4(3x)(2y) 3 (2y) 4 Pre-Calculus 81x 4 3 216x y 2 216x y 2 96xy 3 16y 4 1/31/2007
Review Question
12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … Is the series arithmetic or geometric?
Find the explicit formula Find the recursive formula Find the 100 th term a n = 12 – 2.5(n – 1) a n = a n-1 – 2.5 12 – 2.5(100 – 1) = – 235.5
Find the sum for a 1 through a 100
Pre-Calculus 100 2 1117.5
1/31/2007
Review Question
12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … Is the series arithmetic or geometric?
Find the explicit formula a n = 10(1.2) (n – 1) Find the recursive formula Find the 100 th term a n 10(1.2) 99 = a n-1 ( 2.5) = 690,149,787.7
Find the sum for a 1 through a 100
Pre-Calculus 100 ) 4,140,898,676 1/31/2007
Review Question
Evaluate:
6 ( 1 2
– 43 – ½(3) 2 – ½(4) 2 – ½(5) 2 – ½(6) 2 – ½(9) – ½(16) – ½(25) – ½(36)
Pre-Calculus 1/31/2007
Review Question
Is this sequence arithmetic or geometric?
9, 18, … 144
Pre-Calculus 1/31/2007