Download Lesson #70 – Theoretical and Empirical Probability

L. #68 – How many ways can it happen? Ordering Items
A2.S.10
A2.S.12
Calculate the number of possible permutations (nPr) of n items taken r at a time
Use permutations, combinations, and the Fundamental Principle of Counting to
determine the number of elements in a sample space and a specific subset (event)
To begin our study of probability we will learn how to find the number of ways an event can
happen.
Fundamental Counting Principle: If there are m ways to do one thing, and n ways to do another,
then there are
m  n ways of doing both. The Fundamental Counting Principle is the guiding
rule for finding the number of ways to accomplish two or more tasks.
Ex 1) For example, if a menu offers (A) four different beverages and (B) five different main
dishes there are _____ different meals.
Ex 2) If you want to draw 2 cards from a standard deck of 52 cards without replacing them,
then there are 52 ways to draw the first and 51 ways to draw the second, so there are a total
of 52  51 = 2652 ways to draw the two cards.
n(E) means number of events in E.
The notation is similar to function
notation.
We will call “choosing an outfit” event E. 𝐸1 is choosing a shirt, 𝐸2 is choosing a pair of pants, 𝐸3
is choosing a pair of shoes, and 𝐸4 is choosing a jacket. Given your current wardrobe, how many
different outfits could you make? (Estimate the number of each item in your wardrobe.)
N(𝐸1 )=
N(𝐸2 )=
N(𝐸3 )=
N(𝐸4 )=
N(E)=
Exercise #1: Ten runners are entering in the 100-yard dash. How many ways could the runners
finish first, second, and third?
Exercise #2: If you had twelve songs on your mp3 player, and you set the play mode to random,
how many different ways could you listen to the 12 songs?
~1~
Permutations and Combinations on
the calculator
a) Enter the n value
b) MATH  PRB #2 or #3
c) Enter the r value
A. How many different Saugerties telephone numbers
(246 and 247) can be created?
Remember, permutations are really
just a special application of the
counting principle. In any situation, if
can use a permutation, then you could
B. License plates typically have 6 characters. There
are 26 letters and 10 to choose from. How many
different license plates can be made if no digits or
letters can be repeated and it must begin with four
letters followed by two numbers.
~2~
also use the counting principle. (The
converse is not true. There are
situations where you use the counting
principle but cannot use permutations.
Look at A & B.)
C. Sienna has 6 blocks with one of the letters A, B, C, D, E, and F on each one.
1) In how many ways can she arrange the letters?
2) In how many ways can she arrange the letters if the first and last must be vowels?
Permutations with Repetition
1. How many ways can you arrange the letters in the
word: POINT?
Write out the product for 5 P5 .
How could you rewrite this product
using factorial notation?
2. Write down the arrangements you can make with
the letters in the word, SEE.
In general any permuation in the form
n
Pn  n!
Normally, you would think there would be 3 P3  3!  6 different ways to arrange the letters.
Because of the repeated E’s, some of the arrangements will be the same. Since there are 2 E’s
there are 2 P2  2! repeats. Therefore the total number of arrangements of the letters in the
3!
word SEE is  3 .
2!
Let’s consider the word, RIFFRAFF. There would be
P8
8!

 840 ways to arrange the
4!2!
4 P4 2 P2
8
letters. The 8! represents all of the letters, the 4! represents the repeated F’s, and the 2!
represents the repeated R’s.
3. How many ways can you arrange the letters in the word: KAYAK?
4. How many ways can you arrange the letters in the word: ELEVEN?
5. How many ways can you arrange the letters in the word: DODECAHEDRON?
6. How many ways can you arrange the letters in the word: MATHEMATICS?
~3~
L. #69 – How many ways can it happen? Grouping Items
A2.S.9 Differentiate between situations requiring permutations and those requiring combinations
A2.S.11 Calculate the number of possible combinations (nCr) of n items taken r at a time
A2.S.12
Use permutations, combinations, and the Fundamental Principle of Counting to
determine the number of elements in a sample space and a specific subset (event)
In each of the problems on the first two pages, we were ordering or arranging the objects.
For example, in the race we were not just trying to choose the top three, we wanted to specify
who came in first, second, and third.
When the order does not matter, if we did just want to choose the top three in general, we use
a different measurement called a combination.
n
  is another notation
r 
C
for n r .
5
For example, 5 C3   3  .
 
In essence, a combination is similar to a permutation, but the order of the items does not
matter. Therefore, if we started with a permutation, we would remove all of the situations
where there are the same items in the group but in a different order. Lets look at a simple
example to see how this works. Consider the letters in the word MATH. Below are all of the
different ways that we can arrange three of these letters.
MAT
MAH
MTH
ATH
MTA
MHA
MHT
AHT
AMT
AMH
TMH
TAH
ATM
AHM
THM
THA
TMA
HMA
HMT
HAT
TAM
HAM
HTM
HTA
-Use a permutation to make sure that I did
not miss any of the possible arrangements.
-Now lets assume that the order of the
letters does not matter. In other words, we just want to group three letters together. Cross
out any of the groups above that are repeats. How many groups are you left with?
-Try 4 C3 in your calculator to confirm your answer.
-In is the combination formula, n Cr 
Pr
, what does “r!” do?
r!
n
Exercise #1: Ten runners are entering in the 100-yard dash. How many different groups of
three people could win the top three prizes?
~4~
1. In each situation determine if you would use a permutation or a combination.
a.) Choosing basketball starters.
b.) Choosing a president, vice president, secretary, and treasurer of a club.
c.) Making a password using random numbers.
d.) Selecting a group of marbles from a jar.
2. There are fourteen juniors and twenty-three seniors in the Service Club. The club is to
send four representatives to the State Conference.
a.) How many different ways are there to select a group of four students to attend
the conference?
b.) If the members of the club decide to send two juniors and two seniors, how many
different groupings are possible?
~5~
c.) If the members of the club decides to send one junior and three seniors, how
many different groupings are possible?
d.) If the four chosen members are announced at the conference, in how many
different ways can their names be read?
3. If a three digit number is formed from the digits 1,2,3,4,5,6, and 7, with no
repetitions, tell how many of these three digit numbers will have a number value
between 200 and 500?
4. Adrianna is trying to choose 5 DVD’s to take with her on vacation. She is selecting from
six action movies, eight comedies, and four dramas.
a.) How many sets of 5 DVD’s could she select?
b.) How many sets of 5 DVD’s would contain all comedies?
c.) How many sets of 5 DVD’s would contain two action movies, two comedies, and one
drama?
d.) Once she chooses the 5 DVD’s, in how many different orders can she watch them?
5. How many ways can you arrange the letters in the word: JENNIFER?
~6~
Lesson #70 – Theoretical and Empirical Probability
A2.S.13 Calculate theoretical probabilities, including geometric applications
A2.S.14 Calculate empirical probabilities
Probability is the area of mathematics that answers the question, what is chance that an event
will occur? It can be expressed as a fraction, a decimal, or a percent. The general formula for
the probability of a single event is as follows:
For example, if you have a bag of marbles of which 3 are red, 1 is blue, and 6 are yellow, the
theoretical probability of getting a red marble is:
n(red )
3
P(red) 
 . This probability could be expressed as .3 or 30%.
n(total marbles) 10
We call this probability theoretical because that is what we expect to happen. Theoretically, 3
out of 10 times (or 30% of the time) we should choose a red marble.
Is that always what happens? Do you always get heads when you flip a coin 50% of the time?
Do you always get a 5 when you roll a dice 1/6th of the time?
The actual probability you get when you do an experiment is called the empirical probability.
This makes sense because empirical means “information gained by means of observation,
experience, or experiment.”
How do you think the number of trials, (number of times a person picks a marble) impacts the
relationship between the similarity between the theoretical probability and the empirical
probability?
Law of Large Numbers - as the number of trials of an experiment increases, the empirical
probability approaches the theoretical probability.
A. Think about the marble experiment. What is the theoretical probability you will choose a
purple marble?
B. What is the theoretical probability you will choose a marble and not a stone?
C. What is the theoretical probability you will choose a marble that is NOT red?
~7~
From the three questions on the previous page we learn two important principles:
A. Range for Probability:
B. Probability of the Complement of E (opposite of an event): (notation E’ or E c ).
Ec  1  E
The probability of an event and its complement “complete each other” because their sum is 1 or
100%.
1. The party registration of the voters in Jonesville is shown in the table. If one of the
registered Jonesville voters is selected at random, what is the probability that the
person selected in not a Democrat?
2. Geologists say that the probability of a major earthquake
occurring the San Francisco Bay area in the next 30 years
is about 90%. Is this empirical probability or theoretical
probability?
3. A fair die is tossed. The results appear in the table to
the right.
a) Based on this data, what is the empirical
probability of tossing a 4?
4. What is the theoretical probability of tossing a 4?
Party
# of voters
registered
Democrat
6,000
Republican
5,300
Independent
3,700
Result
Frequency
1
3
2
6
3
4
4
6
5
4
6
7
5. A standard deck of cards contains 52 cards. There are 4 suits: hearts, diamonds, clubs
and spades. Each suit contains 13 cards. You are asked to select a card from the deck
without looking.
b) What is the probability of drawing a ten?
c) What is the probability of drawing a jack OR a diamond?
~8~
d) What is the probability that the card is red AND a king?
e) What is the probability of drawing a black ace OR any face card?
6. A die is tossed. It is a fair, unbiased die. Find the theoretical probability of each
event.
f) A number less than or equal to 3 appears.
g) An odd number is tossed.
h) A number greater than 2 AND even appears.
i) A number less than three OR greater than 4 appears.
7. A letter is chosen at random from a given word. Find the probability that the letter is a
vowel if the word is:
j) ALGEBRA
k) PROBABILITY
l) Explain how you would calculate the empirical probability of randomly choosing a
vowel if each person in the class randomly chose a letter.
Probability and Geometry
Review of area formulas
Rectangle/Parallelogram:
P
desired outcomes desired area

total outcomes
total area
Triangle:
Circle:
8. A 10 x 20 foot mural, depicted below, shows a
triangularly shaped region at the bottom of
the mural. Find the probability that a point
selected at random will lie in this triangular
region of the mural.
~9~
9. A square dartboard is represented on the accompanying
diagram. The entire dartboard is the first quadrant
from x = 0 to 6 and from y = 0 to 6. A triangular region
on the dartboard is enclosed by the graphs of the
equations y = 2, x = 6, and y = x. Find the probability that
a dart that randomly hits the dartboard will land in the
triangular region formed by the three lines.
10. The radius of the bullseye is 1 and the radius of the largest circle is 3. If you threw a
dart, what is the probability of hitting the bullseye, given
that the dart will always land within the boundary of the
outer circle?
11. The accompanying figure is a square. The interior sections are
formed using congruent squares. If this figure is used as a dart
board, what is the probability that the dart will hit the shaded
blue region?
12. Challenge: What is the probability of a dart landing in the
one of the shaded circles if it always hits somewhere on
the square target? Round to the nearest percent.
~ 10 ~
Lesson #71 – Probability with Multiple Events
A2.S.12 Use permutations, combinations, and the Fundamental Principle of Counting to
determine the number of elements in a sample space and a specific subset (event)
Independent Events: An occurrences or outcomes that do not affect each other
Example: Flipping a coin twice. The probability of heads on the 1st flip is independent of the 2nd
flip.
The probability of two or more independent events occuring in a row, one AND then the other,
can be found by multiplying the individual probabilities. (This is similar to the fundamental
counting principle).
AND = multiply
Mutually exclusive events: Two or more events that cannot occur at the same time
Example: Choosing two red marbles OR choosing a red and a green marble. If you choose two
marbles, these events cannot both happen at the same time.
The probability of one mutually exclusive event OR another occurring can be found by adding
their individual probabilities.
OR = add
1. A bag of cookies contains 6 chocolate chip cookies, 5 peanut butter cookies, and 1
oatmeal cookie. Brandon selects a cookie and eats it. Then he selects another. Find the
probability that Brandon selected :
a) 2 chocolate chip cookies (chocolate chip AND chocolate chip)
b) A chocolate chip cookie followed by an oatmeal cookie
c) 2 chocolate chip cookies OR 1 chocolate chip cookie followed by 1 oatmeal cookie
d) Anything but 2 chocolate chip cookies
2. Two colored dice (one red, one white) are rolled. What is the probability of rolling "box
cars" (two sixes)?
~ 11 ~
3. Brianna is using the two spinners shown below to play her new board game. She spins the
arrow on each spinner once. Brianna uses the first spinner to determine how many spaces
to move. She uses the second spinner to determine whether her move from the first
spinner will be forward or backward.
a. Find the probability that Brianna will move
fewer than four spaces backward.
b. Find the probability that Brianna will move one space forward or four spaces
backward.
4. A bag contains 12 red M&Ms, 12 blue M&Ms, and 12 green M&Ms. What is the probability
of drawing two M&Ms of the same color in a row? (When the first M&M is drawn, it is
looked at and eaten.) ________ OR ___________ OR __________
5. A student council has seven officers, of which five are girls and two are boys. If two
officers are chosen at random to attend a meeting with the principal, what is the
probability that one girl and one boy are chosen?
6. Alex's wallet contains four $1 bills, three $5 bills, and one $10 bill. If Alex randomly
removes three bills without replacement, determine the probability that the bills will
total $11.
~ 12 ~
Probability of choosing certain groups
In example 5, the calculation was tricky because you had to consider either a boy being first or
a girl being chosen first. In example 6, you had to consider all of the ways you could select the
bills totaling $11. This would be even more difficult if you had to choose four or more items
where the order of choosing did not matter. There is an easier way. We can use combinations
to count the ways that the group of two student council members or the group of three bills can
be chosen and divide these by the total number of groups in each case.
Go back to examples 5 & 6 and calculate the each probability using combinations.
~ 13 ~
group
Remember, you can only use combinations in probability when the order does not matter. In
other words, you can use it when selecting a group.
~ 14 ~
Lesson #72 – Binomial Probability
A2.S.15 Know and apply the binomial probability formula to events involving the terms exactly,
at least, and at most
In a binomial experiment there are two outcomes, often referred to as "success" and
"failure". If the probability of success is p, the probability of failure is 1 – p. We usually
use the variable q for 1 - p.
For example, if the probability of success, p, is .2, the probability of failure, q, is .8.
Examples of binomial experiments:
• flipping a coin -- heads is success, tails is failure p=_____ q=_____
• rolling a die -- 3 is success, anything else is failure p=_____ q=______
• voting -- votes for candidate A is success, anything else is failure
• determining eye color -- green eyes is success, anything else is failure
• spraying crops -- the insects are killed is success, anything else is failure
Binomial probability problems have the following characteristics:
 There are a number of successive trials
 Each trial has two possible outcomes
 Each trial is independent of the other trials (the outcome of one trial does not
impact the outcomes of the other trials.)
 They words exactly, at least, or at most appear in the problem, usually in italics
(this is not always true, but the regents questions will be set up in this way)
There is a moderately complex formula for calculating binomial probability, but this can also
be done easily on the calculator if you can find three components of the problem: the
number of trials (n), the probability of a single success (p), and the number of desired
successes (r). First, we will work on identifying these components.
n: number of trials
p: probability of success on a single trial
r: number of desired successes
3
chance of winning each game it plays this year.
4
What is the probability it wins exactly three of its first five games?
1. The SHS baseball team has a
~ 15 ~
3
chance of winning each game it plays this year.
4
What is the probability it wins at most three of its first five games?
2. The SHS baseball team has a
3
chance of winning each game it plays this year.
4
What is the probability it wins at least three of its first five games?
3. The SHS baseball team has a
4. On an equatorial island it rains very consistently. The probability of rain on any given
day is 2/3. What is the probability it will rain exactly five days next week?
5. If you drive on 9W through a section of Kingston, there are six traffic lights in a
row. On average, each traffic light is red for 40 seconds, yellow for 5 seconds, and
green for 20 seconds. Find, to the nearest tenth, the probability that you would get
a red light no more than two times.
6. A study shows that 35% of the fish caught in a local lake had high levels of mercury.
Suppose that 10 fish were caught from this lake. Find, to the nearest tenth of a
percent, the probability that at least 8 of the 10 fish caught contained high levels of
mercury
7. On the SAT’s there are five choices per question. On a section of 20 questions, what
is the probability that someone could randomly guess correctly on at least 15 of the
questions?
You will notice that the p-value is usually given in the beginning of the problem while n and r
values are given towards the end of the problem. This is not always the case, but it can
help you decode the question.
Once you can recognize a binomial probability problem and find n, p, and r, you can easily
find the probability on your calculator.
~ 16 ~
nd
Exact Binomial Probability on the Calculator
1. Go to 2 VARS (DISTR)
2. Arrow down and choose binompdf
(This is either 0: or A: depending on your calculator.)
3. Enter the information as follows:
binompdf (number of trials(n) , probability of success(p) ,number of successes(r))
Summary: binompdf (n,p,r)
~ 17 ~
To calculate binomial probability with at most/at least questions, the process is very similar.
Since there are multiple r values, and you want one or the other of them, you want to add
their probabilities together. You can tell your calculator to do so in the following way:
1.
2.
3.
4.
nd
At Most/At Least Binomial Probability on the Calculator
Press 2 STAT (LIST) MATH
Choose 5:Sum
Go to 2nd VARS (DISTR)
Arrow down and choose binompdf
(This is either 0: or A: depending on your calculator.)
5. Enter the information as follows:
binompdf (number of trials(n) , probability of success(p) ,{number of successes(r)})
Summary: sum(binompdf(n,p,{r})
The {} brackets around the r values indicates that all of those numbers are r values. sum(
adds up the individual probability for each r value.
1. Sheffield is batting .267 for the Yankees. What is the probability that he will get at
most 2 hits in his next 7 at bats? Round to the nearest thousandth.
2. The Saugerties wrestling team has won 75% of its matches over the past ten years.
Assuming that their skill level is consistent each year, what is the probability that they
will win at least 5 of their last 7 matches? Round to the nearest percent.
3. If you drive on 9W through a section of Kingston, there are six traffic lights in a row.
On average, each traffic light is red for 40 seconds, yellow for 5 seconds, and green for
20 seconds. Find, to the nearest tenth, the probability that you would get a red light at
most two times.
I can’t find it!
4. On the SAT’s there are five choices per question. On a
section of 20 questions, what is the probability that someone
could randomly guess correctly on at least 15 of the
questions? Express your answer as a decimal and a percent.
~ 18 ~
If you forget where to find
sum or binompdf, all of the
functions on your calculator can
be located in the catalog.
Press 2nd 0 (Catalog) and scroll
through the alphabetized
options. You can press the
button with first letter of the
item you are looking for to get
there faster. The letters are
above the buttons in green.
Lesson #73 - Binomial Expansions
A2.A.36 Apply the binomial theorem to expand a binomial and determine a specific term of
a binomial expansion
In the following expressions, a and b could represent any single term. For example, (2 x  3)3
3
would be the same as (a  b) where a = 2x and b = -3.
1. Simplify (expand): (a  b)2
2. Simplify (expand): (a  b)3
Here is the expansion of (a  b)4
a  b
4
  a  b  a  b  a  b  a  b 
   a  b  a  b    a  b  a  b 
  a 2  ab  ba  b 2   a  b  a  b 
  a 2  2ab  b 2   a  b  a  b 
 a 3  2a 2 b  ab 2

 
  a  b
2
2
3
  ba  2ab  b 
  a 3  3a 2 b  3ab 2  b3   a  b 
 a 4  3a 3b  3a 2 b 2  ab3

 

3
2 2
3
4 
  ba  3a b  3ab  b 
 a 4  4a 3b  6a 2 b 2  4ab3  b 4
3. How long do you think it would take you to find (a  b)5 using that same method?
In the problems above, you were performing a binomial expansion the long way, but a helpful
pattern forms that allows you to perform the expansion of (a  b)n without doing endless
multiplication. This formula or pattern is known as the binomial theorem.
~ 19 ~
The formula for this pattern is actually given to you on the regents. Lets learn how to use it.
(a  b)  C a b  C a b  C a b  ...  C a b
n
n
n
n 1
0
0
n
n2
1
1
n
2
2
0
n
n
n
Notice that a is the first term of the binomial, b is the second term of the binomial, and n is
the exponent on the expansion.
Example: Expand (2 x  5)4 .
1. Identify n, a, and b.
a = 2x
b=5
n=4
2. Plug those values into the formula. It helps to write the terms vertically. Simplify the
resulting expression
( a  b) 
( 2 x  5) 
n
n
Cab 
n
Ca b 
n
Ca b 
n
4
C  2x 5 
4
C  2x 5 
4
C  2x 5 
4
C  2x 5 
4
C  2x 5
4
0
0
n 1
1
1
n2
2
...  C a b
0
n
n
3
3
0
Final Answer:
2
2
1
n
1
1
2
2
0
0
3
 2x  5 
116 x 1 
 4  8 x  5  
 6  4 x  25  
4  2 x 125  
11 625 
4
4
4
3
4
4
3
2
Perform each of the following binomial expansions:
1)
4
4
16 x +
3
160 x +
2
16 x  160 x + 600 x + 1000 x + 625
4
 2x  5 
(4 x  6)3
~ 20 ~
2
600 x +
1000 x +
625
2)
(3x  2 y)5
You will also be asked to identify specific terms in an expansion.
What is the middle term in the example problem?
What is the 2nd term in #1?
When you do not already have the expansion written out and you only have to find one term, you
do not want to write out the entire expansion. Instead, use the formula and the “up, down, up”
pattern to find the term you want.
(a  b)  C a b  C a b  C a b  ...  C a b
n
n
n
0
n 1
0
n
n
Example) What is the 4th term in the expansion of
1. Identify n, a, and b:
n=5
2. Write out the first term.
1st
3. From there, count up (0,1,2,3)
n2
1
1
n
b=4
C0 a n b 0 5 C0 x 5 40
1 4 1
(5,4,3,2) for the exponent on
2 3 2
exponent on b.
n
n
5
for the combination, down
a, and up (0,1,2,3) for the
0
n
 x  4 ?
a=x
term:
2
2
4th term:
4. Simplify the resulting term.
1. What is the middle term of ( x  4 y)6 .
~ 21 ~
5
C3 x 2 43  (10)( x 2 )(64)  640 x 2
2.
What is the 3rd term of the expansion of  3  2i  ?
5
3. What is the last term of the expansion of
( x  2 y)
4. What is the 5th term of the expansion of ( x 2  y 3 )8 .
~ 22 ~
25
.
Lesson #76 – Summations
A2.N.10 Know and apply sigma notation
Our first quote of the year was, “The purpose of mathematics is not to make simple things
complicated, but to make complicated things simple.”
You have seen many examples of this fact throughout the year (multiplication for repeated
addition, exponents for repeated multiplication, function notation, the unit circle, etc.).
Now we will learn a way to express long sums using a compact notation:
A summation is the sum of a group of numbers or expressions. For example, the sum of the
consecutive integers 2 through 7 is ____________________.This sum is easy to write with
words.
Certain problems such as, “the sum of 3n-1 where n is all integers from 2 through 5,” are more
easily understood written in sigma notation. The greek letter, sigma:

is used in math to
abbreviate the summation of a group of numbers.
5
 3n  1 
n2
The letter below sigma identifies the index variable in the expression. This is the variable that
we will plug the numbers into. Any other variables in the problem remain as variables.
The number below sigma gives ___________________ for the index variable.
The number above sigma gives ___________________ for the index variable.
Altogether, this expression tells us to evaluate “3n-1” for all integers 2 through 5, and add the
resulting values.
Practice:
4
1.
a
2
 2a
a 1
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x 

2 
3
 sin 
2.
x1
Mode?
7
3.
 2x  i
x 3
i is the imaginary unit
4.
4
5.
Find the value of
 x 2 y
4 i
i
i 0
i is the index variable
6.
2
3
Find the value of 5  7  (k  1) .
k 0
PEMDAS
7.
Evaluate:
5
 cos  x   2
x 2
8.
If n Cr represents the number of combinations of n items taken r at a time, what is the
3
value of
2 C
r=1
4
r
?
(1) 48
(3) 12
(2) 28
(4) 8
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3
9. Express
 ( x  n)
n
in simplest form:
n 1
At the beginning of the lesson, we started with the reason for summation notation: to write long
sums concisely. The binomial theorem that we learned in the last lesson is a good example of a
long sum. It is also given to you in summation notation on the reference sheet.
(a  b)  C a b  C a b  C a b  ...  C a b
n
n
n
n 1
0
0
n
1
a  b   C a b
n
n
r 0
nr
n
n 2
1
n
2
2
0
n
n
n
r
r
Since summations are so new to you, we will not be using the binomial theorem in this form. I
have found that it only confuses students more at this point because there are so many
variables within the summation notation. I only mention it now so that you can see how much
more concise it is to write it this way.
Summations on the Calculator
These problems can be checked on the calculator as long as the problem does not contain any
other variables other than the index variable. That means all of the problems in this lesson can
be checked this way except for example 5.
To do so, press ALPHA WINDOW (F2). Choose the second option, and enter the problem as you
see it. Go back and try a couple of the example problems.
You still must show your work when doing these problems by writing out the summation. Having
the calculator should help you avoid careless mistakes.
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