6.1 Binomial Theorem (AT 11.5) Name: ________________________________

Obj.: I will be able to read and evaluate binomial coefficients. I will be able to expand binomials using binomial theorem.

VocabularyBinomial Coefficient

Binomial Theorem

Notes Binomial Coefficients

o For _______________________ integers n and r, with n≥ r, the expression (nr ) (read

“_________________”) is called the __________________ _______________________ and is defined by

o The symbol ________ is often used in place of ________ to denote binomial coefficients.

Binomial Theoremo When we write out _____________, where n is a positive integer, a number of

_____________ begin to appear.(a+b )1=a+b (a+b )2=a2+2ab+b2 (a+b )3=a3+3a2b+3ab2+b3 (a+b )4=a4+4 a3b+6a2b2+4ab3+b4 (a+b )5=a5+5a4b+10a3b2+10a2b3+5ab4+b5

o ___________________ form of the binomial expression is a ______________________. Observe the following patterns:

The first term in the expansion of (a+b )n is _____. The exponents _________________ by ____ in each successive term.

The exponents on b in the expression (a+b )n _____________ by 1 in each successive term. In the first term, the exponent on b is _____. The last term is ______.

The ________ of the ____________________ on the variables in any term in the expansion of (a+b )n is equal to _____.

The _________________ ____ ____________ in the polynomial expansion is ________ ________________ than the power of the binomial, n. There are ____________ terms in the expanded form of (a+b )n.

o If we use binomial coefficients and the pattern for the variable part of each term, a formula called the __________________ ______________________ can be used to expand any positive integral power of a binomial:

6.1 Binomial Theorem (AT 11.5) Name: ________________________________

Practice

Evaluate the given binomial coefficients.

1. (131 ) 2. (1003 )

Use the binomial theorem to expand the binomials.

3. (3 x−1 )3 4. (5m+2 )3

5. (6 x− y )5 6. (3a+b )6

Find f ( x−h )−f ( x )

hand simplify.

7. f ( x )=4 x4+1 8. f ( x )=x4+7

6.2 Binomial Expansion (AT 11.5) Name: ________________________________

Obj.: I will be able to find a particular term in a binomial expansion. I will be able to use Pascal’s Triangle to find the coefficient of a term in a binomial expansion.

VocabularyPascal’s Triangle

Notes Finding a ___________________ _________________ in a Binomial Expansion

o The (r+1 )st term of the expansion of (a+b )n is (nr )an−rbr _____________ triangle

o Pascal’s triangle is an array of numbers showing ________________________ of the terms in the expansions of (a+b )n.

6.2 Binomial Expansion (AT 11.5) Name: ________________________________

Practice

Write the first three terms of the binomial expansion, expressing the result in simplified form.

1. ( x+5 )7 2. (m−2 )8

3. (x4−3 y )12 4. (x3+4 )10

Find the indicated term of the binomial expansion.5. 3rd term ; (2 x+ y )8 6. 5th term; (5b−2r )7

Find the term of the binomial expansion containing y127. (x3+ y )21 8. (x4+ y )23

Find the middle term of the binomial expansion.9. ( 4x + x4 )

14 10. ( 7x + x7 )12

6.3 Permutations (AT 11.6) Name: ________________________________

Obj.: I will be able to draw and/or read a tree diagram that describes possible combinations of items. I will be able to use the fundamental counting principle to find the number of choices available. I will be able to find the number of permutations for a set.

VocabularyTree Diagram Fundamental

Counting PrinciplePermutations

Notes Fundamental Counting Principle

o A _________ _____________ is a diagram with _______________ showing the possible ____________________ of items.

o The fundamental counting principle states that the ______________ _____ _______ in which a series of successive things can occur is found by ______________________ the ______________ of ways in which each thing can ___________.

Example: A woman is trying to decide what to wear. She can choose between blue or black pants, a white, yellow, or blue shirt, and black or red shoes. How many different choices of outfit does this woman have?

Tree Diagram:

Fundamental Counting Principle2 pants, 3 shirts, 2 pairs of shoes: ______________________outfits possible.

Permutationso A ___________________ is an _____________ arrangement of items that occurs when:

______ _________ is __________ ___________ than _________. The _____________ of arrangement ____________ ____ __________________

o Permutations of n Things Taken r at a Time The ______________ of possible ______________________ if ____ items are

taken from ____ items is

6.3 Permutations (AT 11.6) Name: ________________________________

Practice

Use the formula for Prn to evaluate the expression.

1. P59 2. P38

Use the Fundamental Counting Principle to solve.

3. The model of the car you are thinking of buying is available in three different colors and four different styles. In how many ways can you order the car?

4. An ice cream store sells 4 drinks, in 3 sizes, and 6 flavors. In how many ways can a customer order a drink?

Use the formula for Prn to solve the following question.

5. A club with twelve members is to choose three officers: president, vice-president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

6. At a benefit concert, ten bands have volunteered to perform but there is only enough time for eight of the bands to play. How many lineups are possible?

6.4 Combinations (AT 11.6) Name: ________________________________

Obj.: I will be able to distinguish between permutations and combinations. I will be able to calculate the number of combinations that are possible for select items from a set.

VocabularyCombination

Notes Combinations

o A __________________ of items occurs when The items are selected from the __________ ________. ____ item is _______ _______ than __________. The ____________ of the items makes _____ _____________________.

o Difference between __________________ and __________________: Permutation – ___________ ______________ Combination – ___________ ______________ ______ ___________________

Formula for Combinationso ________ means the number of combinations of ____ things taken ____ at a time.o The number of __________________ combinations if r items are taken from n items is

o This is the _________ ________________ for the binomial coefficient ______

6.4 Combinations (AT 11.6) Name: ________________________________

Practice

Use the formula for C rn to evaluate the expression.

1. C610 2. C1313

Determine whether the following problem involves a permutation or a combination and explain your answer.

3. A medical researcher needs 30 people to test the effectiveness of an experimental drug. If 78 people have volunteered for the test, in how many ways can 30 people be selected?

4. How many different 2 -letter passwords can be formed from the letters A , B , C , D , E , F , and G if no repetition of letters is allowed?

Use the formula for C rn to solve the following question.

5. You volunteer to help drive children at a charity event to the zoo, but you can fit only 6 of the 19 children present in your van. How many different groups of 6 children can you drive?

6. To win at LOTTO in one state, one must correctly select 7 numbers from a collection of 46 numbers (1 through 46 ). The order in which the selection is made does not matter. How many different selections are possible?

6.4 Combinations (AT 11.6) Name: ________________________________

6.5 Probability (AT 11.7) Name: ________________________________

Obj.: I will be able to calculate empirical and theoretical probabilities. I will be able to determine the probability of an event not occurring.

VocabularyEmpirical Probability

Experiment Sample Space Theoretical Probability

Notes __________________ Probability

o Probabilities of events are expressed as numbers ranging from ______ (or __________) Closer to ______ – event ___________ likely to occur Closer to ______ – event _______ likely to occur

o Empirical probability applies to situation in which we observe how _________________ an event occurs.

The empirical probability of event E, denoted by ________ is

___________________Probabilityo Any occurrence for which the outcome is ______________ is called an

________________. o The set of all possible outcomes of an experiment is the _____________ __________ of

the experiment, denoted by ____.o An event, denoted by ____, is any _________________, or subset, of a sample space.o Theoretical probabilities applies to situations in which the sample space only contains

______________ ___________ __________________, all of which are known. If an event E has ________ equally likely outcomes and its sample space S has

________ equally likely outcomes, the theoretical probability of event E, denoted by _______, is

Probability of an Event Not Occurring

o If we know P(E), the probability of an event E, we can determine the probability that the event will _________ ________________, denoted by ___________.

o Because the sum of the probabilities of all possible outcomes in any situation is ____, the probability that an event E will not occur is equal to ______ ____________ the probability that it _______ _____________.

6.5 Probability (AT 11.7) Name: ________________________________

Practice

6.6 Probability of Multiple Events (AT 11.7) Name: ________________________________

Obj.: I will be able to determine the probability of two events occurring if they are mutually exclusive events, not mutually exclusive events, and/or independent events.

VocabularyMutually Exclusive Events Independent Events

Notes ____ Probabilities with ___________ _______________ Events (Addition Property)

o If it is ________________ for any two events, A and B, to occur __________________________, they are said to be ___________________ _________________.

o If two events are mutually exclusive, the probability that either A or B will occur is determined by adding their ___________________ ______________________.

_______________________________ Set Notation: _______________________

_____ Probabilities That are ______ Mutually Exclusive (Addition Property)o If A and B are events that are not mutually exclusive, the probability that A or B will

occur is determined by _________ their ______________________ ________________ and then __________________ the probability that A and B will occur _________________________.

_______________________________________________ Set Notation: ___________________________________________

_____ Probabilities with ________________________ Events (Multiplication Property)o Two events are independent events if the _______________________ of either of them

has ______ ____________ on the probability of the other.o If two events are independent, we can calculate the probability of the first occurring and

the second occurring by ___________________ their probabilities. _____________________________ Set Notation: _______________________________

6.6 Probability of Multiple Events (AT 11.7) Name: ________________________________

Practice

6.7 Expected Values (M2 13.7) Name: ________________________________

Obj.: I will be able to make random selections and simulate a model. I will be able to determine the expected value for an outcome and use that information to make the best possible decisions. Students will be able to determine fairness.

VocabularyRandom Event Expected Value Fair

Notes Fairness

o ____________ is often a matter of _______________. o A basic game of chance is considered fair if _________ ________________ has an

_____________ __________________ of winning. o A choice is fair if all possible ____________ have an ____________ probability of being

_________. Ex: Two teams decide to play baseball. They want to decide who bats first.

Robert and David are the team captains. They each suggest a method to decide who bats first.

Robert: Flip a coin. If it lands on heads, my team will bat first. If it lands on tails, David’s team bats first.

o __________ _______________ because there is an _________ chance that the coin will land on _______ or ________.

David: Roll a single die. If it lands on 1, 2, or 3, my team bats first. If the roll is 4, 5, or 6, then Robert’s team bats first.

o __________ _______________ because there is an __________ chance of rolling a __________ as there is to roll a __________.

o To help ____________ ________, making _____________ _________________ is a fair way to choose items/people from a set.

Making Random Selectionso You can use probability to make choices and to help make decisions based on

_________ ___________________. o A random event has ___ ________________ ____________ or bias toward one out

come or another. o You can use ___________ __________ _____________ to help you make fair decisions. o Example

There are 28 students in a homeroom. Four students are chosen at random to represent the homeroom on a student committee. How can a random number table be used to fairly choose the students?

Select a _________ from a random number table _________ the line from the table into two digit numbers. Match the ______ ________ numbers less than 28 with the position of

the students’ names on a list. ______________ and numbers greater than 28 are _________________ because they don’t correspond to any student on the list.

6.7 Expected Values (M2 13.7) Name: ________________________________

Making a Simulationo Ex: A cereal company is having a promotion in which 1 of 6 different prizes is given away

with each box. The prizes are equally and randomly distributed in the boxes of cereal. On average, how many boxes of cereal will a customer need to buy in order to get all 6 prizes.

Let the digits from 1 to 6 represent the six prizes. Using a graphing calculator, enter the function ______________ to generate

integers from 1 to 6 to simulate getting each prize. One trial is __________ when ______ ____ ___________ have appeared.

____________ how many boxes of cereal will be bought before all the digits 1 through 6 have appeared.

Conduct additional trials (19 more for ____ total). _____________ the results. Calculating an Expected Value

o _____________ ____________ uses ________________ ___________________ to tell you what you can expect in the long run.

o If you know what ___________ happen mathematically, you will make ____________ ________________ in problem situations.

o The expected value is the ________ of each outcome’s value __________________ by its__________________.

o This is a weighted average.o Using the ______________ ____________ is a matter of selecting the choice with the

____________ expected value.

6.7 Expected Values (M2 13.7) Name: ________________________________

Practice

6.7 Expected Values (M2 13.7) Name: ________________________________

6.8 Discrete Random Variables Name: ________________________________

Obj.: I will be able to use discrete random variables to solve probability problems.

VocabularyDiscrete Random Variable

Support

Notes Random Variables

o Quantities that take on _________________ ___________ depending on __________ or _________________.

o Variables whose values are… ______________ Due to _______________

o Examples # people at a concert # wins of baseball team in a season Height of a student

_______________ Random Variableso Set A is ________________ if either

_____ is a ____________ set such as {1,2,3,4 } It can be put in __________________ correspondence with _____________

numbers (in this case, the set is said to be countably infinite)o A random variable is discrete if its range is a ________________ _________.o Given a random experiment with sample space ____, a random variable ____ is a set

function that assigns one and only one real number to each element ___ that belongs in the sample space ____.

o The set of all possible values of the random variable X, denoted x, is called the _______________, or space, of X.

o NOTE: ____________ ___________ at the end of the alphabet typically represent the ______________ of the random variable. The ___________________ ______________ letters represent the random variable’s ________________ ____________.

6.8 Discrete Random Variables Name: ________________________________

Practice

6.8 Discrete Random Variables Name: ________________________________

6.8 Discrete Random Variables Name: ________________________________