Exam 1 Review
5.1-7.2
Basic Counting Rules- ch. 5
•
•
•
•
SUM rule (for +)
PRODUCT rule (for *)
INCLUSION/EXCLUSION
COMPLEMENT rule
–
number= total – opposite
– Ex: number with at least 2 vowels =
total – (number with 0 or 1 vowels)
5.1, 5.3, 5.5
• Order matters, repetition allowed
– Multiplication Rule
– Ex: Social Security numbers
• Order matters, repetition NOT allowed
– Permutations: P(n,r)= n!/(n-r)!
– Ex: number of ways to pick 1st, 2nd, 3rd from 30
• Order DOESN’T matter, repetition allowed
– section 5.5 (stars and bars; objects and dividers)
– n categories, n-1 dividers, r objects
– C(n-1+r, r) = C(n-1+r, n-1)
• Order DOESN’T matter, repetition NOT allowed
– Combinations: C(n,r)= n!/ [(n-r)!*r!]
– Ex: number of ways to pick a committee of 3 from 30
Binomial
• P(X=k)= nCk * p k q n-k
•
•
= np
σ = (npq)
Basic probability rules
• P(E)=|E|/|S|
• 0<= P(E) <= 1
• P(E ‘ ) = 1 – P(E)
• For Bayes Thm: do tree diagram
• Expected value: E(X)=
• for binomial = np
=
Sample Ch. 5 and 6 problems
• Sample problems:
• 1. Passwords can be comprised of letters or
digits. (uses sum, multiplication, complement
rules)
• How many of them are:
• a) 4-6 characters
• b) 4-5 characters, with exactly 1 digit
• c) 4-5 characters, with exactly 2 digits
• d) 4-5 characters, with at least 2 digits
•
…Probability
• 2. Which "type" of counting problems are these?
(case 1,2,3,4, or 5?)
• a)An ice cream parlor has 28 different flavors, 8
different kinds of sauce, and 12 toppings.
•
i)In how many different ways can a dish of
three scoops of ice cream be made where each
flavor can be used more than once and the order
of the scoops does not matter?
•
ii)How many different kinds of small sundaes
are there if a small sundae contains one scoop of
ice cream, a sauce, and a topping?
…Probability
• b) How many ways are there to choose a dozen donuts
from 20 varieties:
•
i)if there are no two donuts of the same variety?
•
ii)if all donuts are of the same variety?
• iii)if there are no restrictions?
•
iv)if there are at least two varieties?
•
v)if there must be at least six blueberry-filled
donuts?
•
vi)if there can be no more than six blueberry-filled
donuts?
…Probability
• c) A professor writes 20 multiple choice questions, each
with possible answer a,b,c, or d, for a test. If the number of
questions with a,b,c, and d as their answer is 8,3,4, and 5,
respectively, how many different answer keys are possible,
if the questions can be placed in any order?
• d) How many ways are there to assign 24 students to five
faculty advisors?
• e) A witness to a hit and run accident tells the police that
the license plate of the car in the accident, which contains
three letters followed by three digits, starts with the letters
AS and contains both the digits 1 and 2. How many
different license plates can fit this description?
…
• f) There are 7 types of bagels at the store.
• i) How many different ways could you pick 12 of
them and bring them to a meeting?
• ii) How many different ways could you choose to
select bagels to each on 12 consecutive days?
• g)How many ways could we rearrange 13 books
on a bookshelf:
•
i)if all are different?
•
ii)if 4 are identical chemistry books, 6 physics,
and 3 math?
…
• h)How many ways could I there be to select 6
students out of 20 to receive A's?
• i)How many ways could I guess who in this
class will get the best, second, and third score
on the exam?
• j) How many ways can I select 3 women and 3
men from a Math Team (of 20 female
mathletes and 25 male mathletes) to go to the
National Math Tournament?
…
• 3. Number of solutions
• a) How many nonnegative solutions are there
to x1 + x2 + x3 = 30, where x1>1, x2>4, x3>2?
…
• 7. how many bit strings of length 8:
• i) have at least 6 zero‘s?
•
ii) start with 10 and end with 010?
…
• 8. a) How many one-to-one functions exist
from a set with 3 elements to a set with 7
elements (section 5.1)?
Ch. 6: Probability
• Basic Def
• P(E’)
• Bayes
7.1- Recurrence relations example
• Prove: an=n! is a solution to an=n*an-1, a0=1
• Find a solution to
an=n*an-1, a0=1
7.2– Thm. 1
Thm. 1: Let c1, c2 be elements of the real
numbers. Suppose r2-c1r –c2=0 has two
distinct roots r1 and r2,
Then the sequence {a n} is a solution of the
recurrence relation an = ____________
iff an= __________ for n=0, 1, 2… where______
Thm. 2
Thm. 1: Let c1, c2 be elements of the real
numbers. Suppose r2-c1r –c2=0 has ____root ,
Then the sequence {a n} is a solution of the
recurrence relation an = ____________
iff an= __________ for n=0, 1, 2… where______
Summation formula
• Given:
•
=
Solving 2nd degree LHRR-K
For degree 2: the characteristic equation is r2c1r –c2=0 (roots are used to find explicit
formula)
Basic Solution: an=α1r1n+ α2r2n where r1 and r2
are roots of the characteristic equation
Thm. 1 for two roots
Theorem 1: Let c1, c2 be elements of the real
numbers.
Suppose r2-c1r –c2=0 has two distinct roots r1
and r2,
Then the sequence {a n} is a solution of the
recurrence relation an = c1an-1 + c2 an-2
iff an=α1r1n+ α2r2n for n=0, 1, 2… where α1 and α2
are constants.
Thm. 2 for one root
Theorem 2: Let c1, c2 be elements of the real
numbers.
Suppose r2-c1r –c2=0 has only one root r0 ,
Then the sequence {a n} is a solution of the
recurrence relation an = c1an-1 + c2 an-2 iff
an=α1r0n+ α2 n r0n
for n=0, 1, 2… where α1 and α2 are constants.
Ex: 6. an =8an-1 -16an-2 for n≥2; a0=2
and a1=20.
Find characteristic equation
Find solution
Ex: 6. an =8an-1 -16an-2 for n≥2; a0=2
and a1=20.
• Prove the solution you just found is a solution