Download 15.4 – 15.6: probability

Honors Discrete Final Exam Review Guide Fall 2012
CHAPTER 8 - DIGRAPHS
KEY TERMS:
 Arc-Set
 Cycle
 Digraph
 Incident To
 Incident From
 Indegree
 Outdegree
 Path
DIGRAPH: Students should be familiar with the key elements of digraph.
 Find vertices that are incident to or incident from a given vertex.
 Calculate Indegree and Outdegree of vertices.
 Find a path or cycle within a digraph of specific lengths.
 Be able to use an Arc-Set or Digraph to answer the above questions.
A
B
A
B
H
A
B
F
C
G
E
Digraph #1
E
D
D
C
Digraph #2
D
C
Digraph #3
Digraph #4: Consider the Arc-Set {AB, BA, BC, BD, CD, CA, DE, DB, EC, EE}
For Digraphs #1 - #4
1. Find all vertices
a. incident to each vertex.
b. incident from each vertex.
2. Find the indegree for each vertex.
3. Find the outdegree for each vertex.
Digraph #1:
1. Find a path from A to B of
a. Length 4
b. Length 1
c. Length 7
2. Find a path from C to F
3. Find a cycle of length
a. 6
b. 5
c. 3
Digraph #2:
1. Find a path from
a. A to C
b. C to D
c. D to B
2. Find a cycle.
Digraph #3:
1. Find a cycle of length 2
2. Find a cycle that involves vertex A
3. Find a path from A to C
4. Find a path from D B
Digraph #4:
1. Find a path from A to E
2. Find a path from E to B
CHAPTER 9 RECURSIVE AND FIBONACCI SEQUENCES:
Key Terms:
 Fibonacci Sequence
 Binet’s Formula
RECURSIVE SEQUENCES:
 Write the recursive rule and seed
o 16, 12, 8, 4, …
o –11, -2, 7, 16, …
o -3, 6, -12, 24, …
o 5, 15, 45, 135, …
o 5, 11, 23, 47, …
 Find 5th term of the sequence
o an = an-1 + 5; a1 = -2
o an = -3(an-1) + 2; a1 = 4
 Recursive Rule
 Seed
FIBONACCI SEQUENCE:
 Recursive Rule for the Sequence:
FN = FN-1 + FN-2
 Binet’s Formula:
N
N
1 5  
 1   1  5 
 
FN  
 
 


 5   2 
 2  
 Find any term in the Fibonacci
Sequence
 Operations with Fibonacci Numbers
o F26
o F6 + F10
o F18 / F5
o an = 5 - (an-1); a1 = 3
o 2 F10 – 3 F5
o an = an-1 + an-2;
a1 = 5 and a2 = -2
o F4*3
o an = 2(an-1) - (an-2);
a1 = 4 and a2 = 1
o
FF5
o
5 F4
o
2  FF7  F7
o an = 2(an-1) - (an-2);
a1 = 4 and a2 = 1
o an = (an-1)(an-2);
a1 = 2 and a2 = -3
o 2 F20-11
CHAPTER 10 POPULATION GROWTH:
Section 10.2 – Linear Growth
Arithmetic Sequence
 Initial Population, P0
 Common Difference, d
Section 10.3 – Exponential Growth
Geometric Sequence
 Initial Population, P0
 Common Ratio, r
 Recursive Formula: PN = PN-1 + d
 Recursive Formula: PN = r*(PN-1)
 Explicit Formula: PN = P0 + d*N
 Explicit Formula: PN = P0(r)N
SUM of ARITHMETIC SEQUENCE: SUM OF GEOMETRIC SEQUENCE:
A0 + A1 + … + AN-1
a + a(r) + … + a(r)N-1
 Formula:
 Formula:
 A  AN 1 N
SUM  0
2
st
A0 = 1 term being added
AN-1 = Last term being added
N = total number of terms being
added
( r N  1)
SUM  a
( r  1)
a = 1st term being added
r = common ratio
N = total number of terms being
added
Exponential Growth and Decay:
Percent of Change = r (decimal form)
Exponential Growth: PN = P0(1 + r)N
Exponential Decay: PN = P0(1 - r)N
Linear Growth Model:
 Find an explicit formula and 9th generation of the population, P9
1) P0 = 718 and d = 32
3) P5 = 71 and P6 = 83
2) 9.6, 10.05, 10.5, 10.95, …
4) P6 = 65 and P15 = 128
 Find the given sum
5) 12 + 19 + 26 + 33 + 40 + 47 + 54 + 61
8) P0 = 47 and d = 23;
a. P0 + P1 + … + P254
6) P0 = 213 and P29 = 418;
P0 + P1 + P2 + … + P29
b. P50 + P51 + … + P119
7) For 50 terms, 143 + 156 + 169+ …
c. P16 + P17 + … + P39
9) 5 + 17 + 29 + 41 + … + 1157
LINEAR GROWTH WORD PROBLEMS:
10) Starting in the year 2000, the number of crimes committed each year is predicted to
grow according to a linear growth model. During the year 2005, Olympia recorded 97
crimes. During the year 2006, Olympia recorded 112 crimes.
a. How many crimes were committed in 2000?
b. How many crimes are predicted to occur in 2007?
c. How many total crimes were committed between 2000 and 2010?
11) The production of stereos is to be increased by 45 a month. It costs $35 to produce a
stereo. When production started the company spent $280,000 on stereos in January.
a. How many stereos were produced initially?
b. How many stereos will be produced in May?
c. After one year, how many total stereos were produced?
d. After one year, how much money did the company devote to producing stereos?
12) A small business sells $16,500 worth of products during its first year of business.
The owner has set a goal of $1250 in increased sales each year for 30 years. Assuming
the goal is met; find the total sales during the first 12 years this business is in operation.
Exponential Growth Model:
 Find an explicit description and the 9th generation of the population, P9
1) P0 = 14 and r = 3.5
2) 4, 8, 16, …
3) P5 = 99 and P6 = 90
 Find the given sum
4) r = 2.2 and P0 = 3; P0 + P1 + P2 + … + P10
6) r = .5 and P0 = 98304
5) 125 + 150 + 180 + … + 373.248
a. P0 + P1 + P2 + … + P13
b. P7 + P8 + … + P15
EXPONENTIAL GROWTH WORD PROBLEMS:
7) The number of applicants is expected to grow by 23% each year for the next 15 years.
If the original number of applicants was 350, then how many applicants are there
predicted to be in 8 years?
8) The number of reported cases of a virus is supposed to decay by 17% each year for 10
years. There currently are 2,200,000 cases. How many cases are expected 6 in years?
9) The number of certain type of bacteria increases at a rate of 20% every year. Suppose
there were 3600 bacteria in 2009. How many bacteria were there in 2007?
a. Write an equation that describes the number of bacteria per year since 2007.
10) How much interest would you earn on an account with 20% annual interest rate
that you initially invested $2500 after 8 years?
11) A new pair of shoes is marked up 35% and then marked up another 20%. What
percent of the original is the show sold for?
12) A new computer that originally cost $ 1200 is first marked up 20%, then put on sale
15% off, and finally after taxed at 5%. What is the price of the new computer with tax?
15.1 – 15.3 COUNTING THEORY
Key Terms:
 Sample Space
 Random Experiment
 Multiplication Rule
 Sum Rule
 Bit String
 Permutation
 Combination
 Subset
Be able to …
 Solve counting problems that involve
1) Order Matters
2) Order Does not Matter
3) Repetition or Replacement
4) No Repetition or Replacement
5) At Most
6) At Least
7) Complements
 Find the sample space.
 Find the size of the sample space.
 Use the Sum and Multiplication Rule.
Formulas to know:
 PERMUTATION:ORDER MATTERS  COMBINATION: ORDER DOESN’T MATTER
( n)!
n Cr 
r! ( n  r )!
( n)!
n Pr 
( n  r )!
From n total objects permute (order) r of them
From n total objects choose (select) r of them
 Total Number of Subsets of n-elements:
2n or nC0 + nC1 + nC2 + … + nCn-1 + nCn
15.4 – 15.6: PROBABILITY
KEY TERMS:
 EVENT
 COMPLEMENTARY EVENT
 SIMPLE EVENTS
 CERTAIN EVENT




IMPOSSIBLE EVENT
INDEPENDENT EVENTS
ODDS OF
ODDS AGAINST
PROBABILITY:
Probability of an event E occurring:
size of event E
N (E)
Pr(
E
)



N ( S ) size of sample space S
Probability of Complement is the probability of the event E not occurring.
size of event E
N (E)
 Pr( E )  1  N ( S )  1  size of sample space S
C
 Probabilities are always values between 0 and 1
COMPLEMENTARY EVENTS REVIEW: E and EC
 Probabilities: Pr(E) + Pr(EC) = 1
 Size: N(E) + N(EC) = N(S)
Sample Space: S = {σ1, σ2, …, σN }
 σ1, σ2, …, σN represents the simple events (individual outcomes)
Probability Assignment: Pr(σ1), Pr(σ2), …, Pr(σN)




Each of these numbers is between 0 and 1.
Pr(σ1) + Pr(σ2) + …+ Pr(σN) = 1
Pr({}) = 0
Pr(S) = 1
The Multiplication Principle for Independent Events: If events E and F are independent,
then Pr( E then F )  Pr( E )  Pr( F )
BINOMIAL FORMULA:
The probability that an independent event will occur exactly k times out of n is given by
the binomial formula
k
nk
C
p
(1

p
)
n
k
 n = number of trials (tosses of coin, roll of dice, etc)
 k = number of times the event you want will occur
 p = probability event will occur at one trial (probability of coin on single toss or die on
single roll)
 1 – p = probability event will not occur at one trial

n
C k = choosing k of the n trials to be the event you want
ODDS:
For an arbitrary event, odds represent a comparison of the number of ways than even can
occur (favorable outcomes) versus the number of ways an event does not occur
(unfavorable outcomes).
 Favorable Outcomes = N(E)
 Unfavorable Outcomes = N(Ec)
 The odds of (odds in favor of) the event E are given by the ratio N(E) to N(Ec).
 The odds against the event E are given by the ratio N(Ec) to N(E).
Make sure to simplify all odds by the greatest common factor.
Be able to…
(1) calculate odds
(2) convert probabilities to odds AND convert odds to probability
CHAPTER 15 PRACTICE PROBLEMS
1) A license plate consists of 3 capital letters (A through Z) and any 4 single digit
numbers (0 through 9).
a. How many different license plates are possible?
b. How many different license plates contain no repeated characters?
c. How many different license plates start with an M and end with an odd
digit?
2) A password consists of 5 characters. In the password a character is either any
letter or single digit number, and the letters ARE NOT case sensitive. How many
passwords…
a. are possible if there are no
c. do not contain a repeated character?
restrictions?
b. are only letters?
d. contain exactly two A’s and three 4’s?
3) A password consists of 7 characters. In the password a character is either any
letter or single digit number, and the letters ARE case sensitive.
a. How many passwords are possible if there are no restrictions?
b. How many passwords are only numbers
c. How many passwords do not contain odd numbers?
d. How many passwords start with a letter and the last character is a number?
4) How many five-digit numbers (0 to 99,999) …
a. are odd?
c. end in an 8?
b. divisible by 5?
d. have no repeated digits?
5) A fair coin is flipped 6 times and record whether the coin is heads or tails.
a. What is the size of the sample
c. What is the probability of at least
space?
2 heads?
b. What is the probability of exactly
d. What is the probability of at most
4 heads?
3 tails?
6) A die is rolled 8 times and it is recorded what number is shown on each roll.
a. What is the size of the sample space?
b. What is the probability of rolling exactly three 6’s?
c. What is the probability of rolling exactly five even numbers?
7) Draw 2 card from a standard deck of 52 WITHOUT replacement
a. What is the probability to draw 2 kings?
b. What is the probability to draw a jack and ace?
c. What is the probability of a flush (2 of the same suit)?
8) Draw 2 card from a standard deck of 52 WITH replacement.
a. What is the probability to draw a pair (2 of the same value)?
b. What is the probability of a red then black?
c. What is the probability of a red and black?
9) For each of the following probabilities, calculate the odds for and the odds
against each event occurring.
a. Pr(A) = 0.12
b. Pr(B) = 0.17
c. Pr(C) = 0.24
d. Pr(D) = 0.15
10) Find the probability of event E for each of the given odds.
a. The odds of E are 7 to 13.
c. The odds of E are 11 to 9.
b. The odds against E are 47 to 3.
d. The odds against E are 7 to 1