Notes #4-___
Date:______
Multipleevents.
Tree diagram
9.1Basic Combinatorics (669)
Discrete math: distinct values, not continuous (daily attend)
I. Simple Counting Problems –
You have a bag with a penny, dime, nickel and quarter.
How manyways can the sum be greater than 10¢ if…
A.With Replacement: you select a coin, write down the
value and put the coin back in the bag. You pull out a
coin again and add the value to the first.
1,11,51,101,25
5,15,55,105,25
10,110,510,1010,25
25,125,525,1025,2512 ways
B.Without replacement: you select a coin and then
select another coin (or you select two at once)?
1,55,110,125,1
1,105,1010,525,5
1,255,2510,2525,1010 ways
II.Fundamental Counting Principle – the number of ways
that multiple events can occur is the product of the
outcomes (number of elements) of each those events.
Example: Your mom says you can have a pb & j, ham or
tuna sandwich on white or wheat with milk, oj or Pepsi.
How many different lunches could you have?
One event.
Factorial!
Define: 0! = 1
Also: P(n, r)
You would expect 4!?
Ex.1How many license plates can be made with:
a) 3 different letters & 4 different non-zero digits
b) 3 letters and 4 digits (0-9)
III.Permutation – an ordering (ranking) of outcomes.
Example: How many ways can you arrange a family of
fivein arow for a photograph?
______
A.How many ways can you arrange any two of them?
______Which leads to nPr = .
Ex.2How many ways can 8 runners be awarded gold,
silver and bronze medals?
B.Distinguishable (distinct) Permutations
Example: Arrange the letters A, A, B, C.
#1AABC then switch the As and you get #2 AABC.
How many ways can the As be arranged? ______2!
So the formula is specifically or generally .
Ex.3Distinguishable arrangements of:
a) TENNESSEEb) GREENEGGS
IV.Combinations – a selection of outcomes.
Example: How many different 2 topping, small, deep
dish pizzas can we get if there are 10 toppings?
nCr = . Why the extra r! compared to nPr?
Ex.4How many ways can you choose a committee of 5
members from a student government of 9 girls & 6
boys? What if there had to be 3 girls and 2 guys?
Ex.5How many ways can you select 4 of your dad’s 20
CDs for a road trip? In how many ways could you
listen to four of the 20 CDs on the trip?
Deck of Cards:In each suit:
4 suits (2 red, 2 black)*3 face cards (J, Q & K)
Club, Diamond, Heart, Spade*4 letters (J, Q, K & A)
20 even & 16 odd #s*9 #s (2-10)
Ex.6How many ways can you have:
a)2 red face cards?b)3 even cards?
Factorial Notation: n! = 1 · 2 · 3 ··· n
0! = 1 (this value is defined as such)
4! = 1 · 2 · 3· 4 = 24
(n + 1)! = 1 · 2 · 3 ··· (n - 1) · n · (n + 1)
Ex.7Evaluate:
a)b)
c)
d)
e)
Notes #4-___
Date:______
9.2The Binomial Theorem (678)
Expand:
(x + 1)0 =1
(x + 1)1 =1x + 1
(x + 1)2 = 1x2 + 2x + 1
(x + 1)3 = 1x3 + 3x2 + 3x + 1
Pascal’s Δ:
Compare to:
nCr (0, 0) =
nCr (1, 0) & nCr (1, 1) =
nCr (2, {0, 1, 2}) =
nCr (3, {0, 1, 2, 3}) =
nCr (4, {0, 1, 2, 3, 4}) = or y = nCr (4, x) and use the table.
Combination (Binomial coefficient): nCr =
Ex.1Evaluate 8C3 by hand.
Ex.2Expand and simplify
a)(x – 3)4 =
x4 – 12x3 + 54x2 – 108x + 81
b)(2x – 3y)5 =
32x5 – 240x4y + 720x3y2 – 1080x2y3 + 810 xy4 – 243y5
Ex.3Find the fifth term:
a)(2x + 1)9b)(x – 2)13
4032x513C4(x)9(-2)4 = 11440x9
Ex.4Find the coefficient of the term with a7 in the
expansion of (a – 3b)10.
-3,240
Ex.5Use (b + g)8 to find the probability of having 4 boys
and 4 girls in a family of 8 children.
9! = 9(9 – 1)! or 9(8!)
n! = n(n – 1)! for n 1(n + 1)! = (n + 1)n! for n 0
Ex.6Prove that
Notes #4-___
Date:______
9.3 Probability (683)
Probability = , P(E) =
0% Impossible 0 P(E) 1 Certain 100%
Ex1)What is the probability of rolling an even # on a
six-sided die?
Outcomes in event: 2, 4 & 6(# outcomes)
Outcomes in sample space: 1, 2, 3, 4, 5 & 6(# outcomes)
Ex2)What is the probability of rolling a sum that is prime
on a single roll of two fair six-sided dice?
Deck of 52 Cards:In each suit:
4 suits (2 red, 2 black)*3 face cards (J, Q & K)
Club, Diamond, Heart, Spade*4 letters (J, Q, K & A)
20 even & 16 odd #s*9 #s (2-10)
Ex3)What is the probability of drawing:
a)2 red face cards?
b)3 even cards?
Mutually exclusive: 2 events that have no common outcomes.
If A ∩ B = Ø then P(A or B) = P(A) + P(B).
Ex4)What is the probability of drawing:
a)a black 7 or a heart?b)even # or a king?
Not mutually exclusive: P(A or B) = P(A) + P(B) – P(A ∩ B)
Ex5)What is the probability of drawing:
a)a queen or a club?b)red card or a six?
Complementary Event: P(E) + P(E') = 1 so P(E') = 1 – P(E).
Ex6)What is the probability of not drawing:
a)a queen or a club?b)red card or a six?
Ex7)48% of the students at a school are girls and half of
them play sports. 51% of all the students play sports.
a) What % of the students who play sports are boys?
b)If a student is chosen at random, what is the
probability that he is a boy who doesn’t play sports.
Conditional Probability: the probability of an event that
depends on an earlier event.
Ex8)A shirt is drawn at random from one of two
identical drawers (drawer A has 3 t-shirts & 2
sweatshirts and drawer B has 2 t-shirts). What is the
probability that a t-shirt was drawn from drawer A?
Binomial Distribution:
P(E): Probability event happens
[P(E) + P(E')]nP(E'): Probability it doesn’t happen
n: the # of trials
Ex10)10% of African-Americans are carriers of the genetic
disease sickle-cell anemia. Find the P(of # carriers) in
a sample of 20 African Americans:
a) P(3)b)P(at most 2)
20C17(.1)3(.9)1720C18(.1)2(.9)18 +
20C19(.1)1(.9)19 + (.9)20
Notes #4-___
Date:______
9.4 Day 1: Sequences (696)
Sequence: Ordered list of numbers (ranked list): a1, a2, a3…
Number / 1st / 2 nd / 3 rd / n thTerm / a1 / a2 / a3 / an
Said / “a” sub 1 / “a” sub 2 / “a” sub 3 / “a” sub n
Finite SequenceInfiniteSequence
Terms: 1 , 3 , 5 , 7 2, 5, 8, 11, …
The subscript represents the number's place in the list:
a3 (a sub 3) is the third # in our list i.e. a3 = 5.
Ex.1Find the first five terms of the sequence given by:
an = 5 + 2n(-1)n.
Ex.2Write an expression for the apparent nth term of the
sequence:
Recursive definition: given initial term(s), terms are then
defined using the previous term.
Ex.3a1 = -11
an = an-1 + 5* an-1is the number before an
Write the first five terms and find a100.
Ex.4Write the first five terms:
a1 = 3an = 2·an-1
Limits of Infinite Sequences
{an} = a1, a2,a3,a4, … an
If thean = a finite L then the sequence converges and L is the limit of the sequence. Otherwise it diverges.
Ex.5Does the sequence converge? If so, find the limit.
a)b)
c)0.1, 0.2, 0.3, 0.4, …d)10n – 10
e)f)5n
A number (term : an) in an arithmetic sequence is equal to the number before it (an-1) plus the common difference (d).
d1 = a2 – a1 d2 = a3 – a2 dn = an – an-1
Ex.65 , 11 , 17 …Recursive definition:
a1 = 5
an = an-1 + 6
Explicit Formula: an = a1 + (n – 1)·d
Ex.7Find a17 for -3, 4, 11, 18 …
Ex.8In an arithmetic sequence, a3 = 14 and a8 = 44, write the
first five terms and a formula.
a8 = a3 + 5d
Geometric Sequences
3, 6, 12, 24, 48, …#1. Arithmetic? d1 & d2 = ?
#2. Geometric? r1 & r2 = ?
Common ratio:r =
Ex.9Find a10 in 3, 6, 12, 24, 48, …
Ex.10Find a formula for an and a10 for
a)1, -1, 1, -1…b)4, 2, 1 …
What comes next in the pattern?
#1)Fibonacci Sequence: 1, 1, 2, 3, 5, 8 …
#2)31, 28, 31, 30, …31 {Days in the months}
#3)J, F, M, A, …M {Names of Months}
#4)3, 3, 5, 4, 4, …3 {# of letters in #s}
#5)Z, O, T, T, F, F, …S {Whole numbers}
#6)A, E, F, H, …I {straight letters}
#7)8, 5, 4, 9, 1, …7 {Alphabetically}
#8)7, 8, 5, 5, 3, 4, …4 {# of letters in months}
#9)S, M, T, W, …T {Days of the Week}
#10)S, E, Q, U, …E {the word sequence}
Notes #4-___
Date:______
Karl F. Gauss
(1777-1855)
9.4 Series (701)
Series: the sum of a list of terms: S5 = 2 + 5 + 8 + 11 + 14 = ?
Summation (Sigma) Notation
= a1 + a2 + a3 + … + an(i: index of summation)
Ex.1Find the sum:
a)b)
c)d)e)
Ex.2Find the sum of the first 100 natural numbers.
1 + 2 + 3 + … + 98 + 99 + 100 = S100
100 + 99 + 98 + …+ 3 + 2 + 1 = S100
101 + 101 + 101 + … + 101 + 101 + 101 =2S100
100(101) = 2S100
Sn = a1 + (a1+d) + (a1+2d) +…+ (an – 2d) + (an – d) + an
+Sn = an + (an – d)+ (an – 2d) +…+ (a1+2d) + (a1+d) + a1
2Sn = (a1 +an) + (a1 +an) +…+ (a1 +an) + (a1 +an)+ (a1 +an)
2Sn = n(a1 +an)
Sn = Sn =
Ex.3Find S100 and write in sigma notation: 5, 8, 11, 14…
Ex.4A theater has 30 seats in the 1st row and 2 more in each
subsequent row. How many seats are there if there are
78 seats in the last row?
Ex.5Find a10 in 3, 6, 12, 24, 48, …
Ex.6Find a formula for an and a10 for
a)1, -1, 1, -1…b)4, 2, 1
Derive the Geometric sum formula:
Sn = a1 + a1·r + a1·r2 + … + a1·rn-2 + a1·rn-1
r·Sn = a1·r + a1·r2 + … + a1·rn-2 + a1·rn-1 + a1·rn
=
Ex.7Find S10 for the sequences in Ex.5 & Ex.6.
Ex.8a4 = 54 & a7 = 1458, find S7 if it is a geometric series.
If │r│ < 1 then the series converges:
Ex.9Find S∞ for 4 + 2 + 1 + …
Ex.10Find the sum:
a)b)
Ex.11Convert the repeating decimal to fraction form.
a).797979…b)-3.14141414…
.79 + .0079 + ...
a1 = .79 & r = .01
Formulas:
SequencesSeries
an = a1 + (n – 1)·dSn =
an = a1 · (r)n-1
Notes #4-___
Date:______
There is a formula on (746) that we also used in 3.6, but the geometric sum is better.
9.5 Mathematical Induction (711)
Recall: Formula for compound interest:
A:the balance
P:the principle
r:annual interest rate (as a decimal)
n:compounded this many times (quarterly, monthly..)
t:time in years
Ex.1How much money will you have in 6 years if you
invest $20 a month @ 5% compounded monthly.
n = 12 · 6, a1 = 20 and
S72 = 20 + 20(1.004) +…+ 20(1.004)71 + 20(1.004)72
FV = S72 = =
Ex.2How much money will you have in 10 years if you
invest $50 quarterly @ 7% compounded quarterly?
Step #1: Show that P1 is true.
Step #2: Show that for any positive integer k, if Pk is true,
then Pk+1 is also true. (For sums: Sk+1 = Sk + ak+1)
Prove by Induction:
Ex.3Sn = 5 + 7 + 9 + 11 + … + (3 + 2n) = n(n + 4)
S1 = 1(1 + 4) = 1(5) = 5 True
If Sk = k(k + 4) is true, then...
Sk+1 = (k + 1)[(k + 1) + 4] = (k + 1)(k + 5) = k2 +6k + 5
Sk= k(k + 4) and ak + 1=[3 + 2(k + 1)] = 3 + 2k + 2 = 2k + 5
k2 +4k+ 2k + 5 = k2 +6k + 5
Ex.4
S1 = = = True
If Sk = is true, then... Sk+1 =
Sk + ak + 1= + = + =
+ =
Ex.55 is a factor of (42n – 1) or (42n – 1) is divisible by 5.
P1: 42(1) – 1 = 15 which is 5(3) True
If Pk: 42k – 1 = 5r then Pk+1: 42(k+1) – 1 = 5s
42(42k – 1) = 425r42k+2 – 16 = 80r
42k+2 – 16 + 15 = 80r + 15
42(k+1) – 1 = 5(16r + 3)
42(k+1) – 1 = 5s
Ex.65n – 1 is divisible by 2 for all positive integers n.
P1: 51 – 1 = 5 – 1 = 4 which is divisible by 2. True
If Pk: 5k – 1 = 2r then Pk+1: 5(k+1) – 1 = 2s
5(5k – 1) = 52r5k+1 – 5 = 10r
5k+1 – 5 + 4 = 10r + 4
5k+1 – 1 = 2(5r + 2)5k+1 – 1 = 2s
Ex.7n2 > 2n, n 3
P3 = 32 > 2(3) or 9 > 6 which is true.
If Pk: k2 > 2k then Pk+1: (k + 1)2 > 2(k + 1)
k2 + 2k + 1 > 2k + 2k2 > 1
On the TI-89:
Math
3: List
1: seq(expression, variable, low, high)
6: sum({a1, a2,a3, … an}) or sum(seq(expression…))
Ex.7Find the first 5 terms of 3n! ÷ n2.
Graph
4: Sequence
Y=
u1= u1(n – 1) + 2an = an-1 + 2
ui1= 7a1 = 7
Ex.8Graph: a1 = 3an = 2·an-1
an = a1 + (n – 1)·dSn =
an = a1 · (r)n-1
Ex.2= 12 + 22 + 32 + 42 + … + n2 =
23. = 1 + 2 + 3 + 4 + … + n = =
25. = 13 + 23 + 33 + 43 + … + n3 =
Notes #4-___
Date:______
9.6 Statistics and Data - Graphical (717)
Statistics: the science of data (discrete or continuous)
Individual: usually people, objects, transactions or events
that form a population (set) we wish to study.
Variable: characteristic or property of an individual
Categorical (qualitative): variables that cannot be
measured on a natural numerical scale; they can only be
put into groups.
Quantitative: measurements that are recorded on a
naturally occurring numerical scale.
Ex.1Classify each variable as Categorical or Quantitative:
Unemployment rates of the 50 states
Size of a car (compact, mid-size etc.)
Political PartyTemperature
A taste-tester’s ranking
River where each fish was captured
DDT concentrationSpecies
LengthWeight
CategoricalQuantitative
Size of carUnemployment rate
Political PartyTemperature
Taste-tester rankingDDT concentration
RiverLength
SpeciesWeight
How do we find the angles?
Ex.2Construct a pie chart and a bar graph for each year.
Is smoking a cause of lung cancer? / 1954 / 1999Yes / 41% / 92%
No / 31% / 6%
No opinion / 28% / 2%
Stemplots (Stem-and-leaf)
A stem-and-leaf plot arranges data by separating the digits into stems (the beginning digits) & leaves (the ending digits).
Ex.3Make a stemplot for the % of student loans in default.
AZ 12.1NV 10.1 60 4
CA 11.4NM 7.5 71 5 9
CO 9.5OK 11.2 84
HI 12.8OR 7.9 95
ID 7.1UT 6.0101
MT 6.4WA 8.4112 4
121 8
Arizona / 22% / 520California / 44% / 484
Colorado / 29% / 511
Idaho / 16% / 501
Nevada / 24% / 486
New Mexico / 12% / 524
Oregon / 50% / 486
Texas / 45% / 462
Utah / 6% / 536
Washington / 37% / 494
What is lost in a frequency table?
Split Stem Stemplots: Stems split to spread out the data.
Ex.4Make a split stemplot for the SAT scores from 2000.
CA1015NH1039
CT1017NJ1011
HI1007NY1000
ME1004RI 1005
MD1016VT1021
MA1024VA1009
NV1027WV1037
Back-to-back Stemplots
Ex.5The table lists the % of graduates taking the SAT and
their average Math score. Create a back-to-back
stemplot with the states with less than 25% on the left.
Frequency Tables: the # of occurrences per interval.
Ex.6Create frequency tables for the data in Ex.4 & 5.
How is this different from a bar graph?
Be sure to distinguish between the two sets of data.
Summary:
Histogram: displays the info of a frequency table.
Ex.7Create histograms for the data in Ex.5.
Time Plots:a line graph (not linear) where discrete points are
connected to reveal trends in data over time.
Ex.8Make time plots for the data below on the same set of
axes. Which decade did the largest decrease occur for
each? Give a possible explanation.
Median Age atFirst Marriage
Year / Male / Female
1900 / 25.9 / 21.9
1910 / 25.1 / 21.6
1920 / 24.6 / 21.2
1930 / 24.3 / 21.3
1940 / 24.3 / 21.5
1950 / 22.8 / 20.3
1960 / 22.8 / 20.3
1970 / 23.2 / 20.8
1980 / 24.7 / 22.0
1990 / 26.1 / 23.9
2000 / 26.8 / 25.1
Notes #4-___
Date: ______
A statistic is resistant if it is not strongly affected by outliers.
Are these means in Ex.1 resistant?
9.7 Statistics and Data - Algebraic (730)
Statistics:the science of data. Collecting, classifying,
summarizing, organizing, analyzing and
interpreting numerical information
Measures of Central Tendency (describes “normal”):
Given a set of numbers {x1, x2, …, xn} the
* Meanis
* Medianis the middle number if n is odd.
or the mean of the two middle #s if n is even.
* Mode the #s that occur most (may be more than one).
Ex.1The mean age of 4 people in a car is 20. Imagine
the people in your head. What do you see?
A 70 year old grandma and her grandkids at 2, 3 & 5?
Or four 20 year olds cruising?
The mean yearly income for 5 people is $100,000.
Imagine the people in your head. What do you see?
Did you imagine a boss making $400,000 and her four
employees at $25,000? Or did you imagine 5 partners
making equal shares?
Ex.2Find the mean, median and mode for Test #4-1 (S07).
47, 38, 14, 26, 33, 39, 67, 39, 16, 44, 44, 40, 33, 45,
36, 55, 28, 53, 43, 43, 35.
Do not confuse with the range of a f(x).
Q1 Lower quartile QL
Q2 Middle quart. QM
Q3 Upper quartile QU
Range: maximum – minimum
Ex.3Find the range of Ex.2.
Ex.4Find the mean, median, mode and range:
Quiz Scores for a classScore / 10 / 9 / 8 / 7 / 6 / 5 / 4 / 3 / 2 / 1 / 0
Frequency / 4 / 4 / 4 / 4 / 2 / 2 / 5 / 3 / 0 / 2 / 2
Mean: [10(4) + 9(4) + … + 0(2)] / 32 = 189 / 32 ≈ 5.91
Median: 6.5 (16th # is a 7 and the 17th # is a 6 so 13/2 = 6.5)
Mode: 4 occurs the most
Range: 10 – 0 = 10
A boxplot (box-and-whisker plot) divides data into quartiles (4 parts). The 2nd quartile is the median. The 1st quartile is the median of the data less than the 2nd quartile and the 3rd is the median of the data greater than the median.
The five-number summary {min, Q1, median, Q3, max}.
Ex.5Draw a boxplot for Test #4-1 (S07).
47, 38, 14, 26, 33, 39, 67, 39, 16, 44, 44, 40, 33, 45,
36, 55, 28, 53, 43, 43, 35.
The interquartile range IQR = Q3 – Q1. The IQR gives the range of the middle half of the data and is more resistant than the range.
Outlier:a number in a data set that is more than 1.5(IQR)
below Q1 or above Q3.
Standard deviation is strongly affected by outliers.
You don’t need to memorize these .
Summary:
A modified boxplot separates outliers as isolated points.
Ex.6Make a boxplot for the % of student loans in default.
AK19.7NV 10.1
AZ 12.1NM 7.5
CA 11.4OK 11.2
CO 9.5OR 7.9
HI 12.8UT 6.0
ID 7.1WA 8.4
MT 6.4WY 2.7
Standard Deviation and Variance (788)
Typically, two-thirds of the data values are within one standard deviation (s when calculated from a sample or σ if the entire population was used) of the mean. s will be larger.
Variance is σ2.
Ex.7Find the standard deviation and the variance for Ex.6.
APPS
Stats/List E… (Enter)
Enter data into list1
F4 (Calculate)
1:1-Var Stats… (Enter)