algebra 2/trig: chapter 16 probability - white plains … algebra 2/trig: chapter 16 probability in...

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Algebra 2/Trig: Chapter 16 Probability In this unit, we will…

Determine whether a situation is modeled by a permutation or a combination, or a factorial

Generate and use Pascal’s Triangle to determine combinations

Determine exact probabilities that are modeled by Bernoulli trials

Determine “at least” and “at most” probabilities that are modeled by Bernoulli trials

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Table of Contents

Day 1: Permutations and Combinations SWBAT: Determine whether a situation is modeled by a permutation, combination, or a factorial Pgs. 3 – 8 in Packet

HW: Pgs. 9 – 10 in Packet

Day 2: Probability

SWBAT: Calculate the probability of an event

Pgs. 11 – 16 in Packet


QUIZ on Day 3 ~ 7 min

Day 3: Binomial Theorem

SWBAT: Expand a binomial using the Pascal’s Triangle/Binomial Theorem Pgs. 19 – 23 in Packet


Day 4: Bernoulli Trials with Exactly

SWBAT: Determine exact probabilities that are modeled by Bernoulli trials Pgs. 26 – 29 in Packet


Day 5: Bernoulli Trials with At Least and At Most

SWBAT: Determine “at least” and “at most” probabilities that are modeled by Bernoulli trials Pgs. 33 – 36 in Packet


QUIZ on Day 6 ~ 7 min

Day 6: Practice Test

SWBAT: Solve problems involving Probability, Permutations, Combinations, and Bernoulli Trials


Day 7: Test

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Day 1: Algebra2/Trig: Chapter 16-2: Permutations and Combinations Let’s assume there are 25 students in this class. If I were to pick a President, a Vice-President, and a Secretary of the class AT RANDOM,

= _____________________ total ways. # ways to choose a

P

#ways to choose

VP

#ways to choose S

A permutation is an arrangement of people or things where the ORDER MATTERS. The notation nPr means you are taking n objects and you are arranging r of them. When you’re writing a permutation, you start with the “n,” and count down towards 1, multiplying the whole time, r number of times.

nPr = ( ) ( ) ( )

r times

Meaning The notation The math Arranging 4 objects from 10 10P4

Arranging 2 objects from 52


The last case is a special case of a permutation, where you are arranging ALL of the objects. In nPr, if n=r, this is called a factorial. Factorial is represented by an exclamation point.

10! = 10P10 = = a really big number.

Meaning The notation The math Arranging 5 objects from 5 5P5 or 5!


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A combination is group of people or things where the ORDER DOESN’T MATTER.

You don’t have to memorize that formula… your calculator will do all of it! Your calculator can do all of this from the MATH PRB menu. Example: 10P4 type 10 MATH PRB 2 ENTER 4 ENTER Example: 5! type 5 MATH PRB 4 ENTER 4 ENTER Example: 10C4 type 10 MATH PRB 3 ENTER 4 ENTER

Type Words to Look For Sample Problem Answer

Permutation (Lining up different objects)

Line up Arrangement Put in Order Rank

# of ways to line up 3 different objects out of 10

10P3 = = 720

Factorial Same as a permutation, but you’re lining up or ordering ALL of the objects.

Same as permutation

# of ways to line up all 5 objects out of 5

5P5 = 5! =

Combination (taking a group of objects)

Group Committee Team

# of ways to group 3 objects out of 10

10C3 =

=120

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Problem Permutation or Combination?

Write in nPr, nCr, or n! notation and use calculator to answer

1. The # of ways to line up 10 different chairs

2. The # of ways to line up 4 out of 10 different chairs

3. The # of ways to pick a group of 3 chairs from 10

4. # of ways to pick a president, a VP, and a treasurer from the students in this class.

5. # of ways to pick a committee of 3 people from this class

6. # of ways to line up all the people in this class.

7. # of ways to choose a 3-scoop sundae from 31 flavors of Baskin Robbins

8. # of ways to make a 3-scoop cone from 31 flavors if you care what order the scoops are in

Special Case: Probability with repetition: Example: How many ways are there to arrange the letters of the word “Mississippi?” Every time there is a letter that repeats, count how many times each letter repeats, divide the answer by that number factorial. Solution: There are 11 letters in “Mississippi”. The “I”s repeat 4 times, the “s” repeats 4 times and the “p” repeats twice.

1. 2.

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Passcodes 3. 4. Arrangements 5. In how many different ways can the letters of the word PENCIL be arranged if the first

letter must be a consonant? 6. In how many different ways can the letters of the word PENCIL be arranged if the first

and last letter must be a consonant?

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Combinations with “And”

When you see the word “And” this means to __________________ 7. 8.

9.

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CHALLENGE:

SUMMARY Exit Ticket

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Day 2: Algebra2/Trig: Permutations and Combinations Review

Recall from Integrated Algebra:

Probability of an Event = Number of successes

Total possible outcomes

Probability is expressed as a fraction/decimal between 0 and 1.

Probability can be expressed as a percent between 0 and 100.

Absolute certainty has a probability of 1.

Total impossibility has a probability of 0.

The sum of all probabilites in a situation is = 1.

The probability of an event is written as P(E).

Ex: A box of golf balls contains three yellow, four white, and one red.

If Lee chooses one golf ball from the box without looking, find the probability

that the golf ball is

a. yellow b. red c. not red d. green

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3. You roll two six-sided dice. What's the probability of not rolling doubles?

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Combinations and Probability

Example 1:

Example 2:

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Example 3: There are 3 seniors and 15 juniors in Mr. Cameron’s math class. Three students are chosen

at random from the class. What is the probability that the group consists of 1 senior and 2 juniors?

Example 4: A committee of 5 is to be chosen from a group of 8 men and 6 women. What is the

probability that the committee formed consists of 2 men and 3 women?

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SUMMARY Exit Ticket

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Day 3 - Algebra2/Trig: Pascal’s Triangle and Binomial Expansion

( ) There is a pattern in the a’s. There is a pattern in the b’s. There is a pattern in the coefficients.

( )

( ) ( ) ( ) ( )

To expand a binomial of the form ( )

Write out powers of a. The powers of a start at ____ and ___________.

Write out powers of b. The powers of b start at ____ and ___________.

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Pascal’s Triangle

Write out the coefficients. These come from Pascals’s triangle, OR you can

use combinations starting at ____C___.

Example 1: Set up the binomial expansion for (2a – b)3 by using the binomial expansion theorem.

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Example 2: Set up the binomial expansion for (4x – 3)5 by using the binomial expansion theorem.

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2.

3. What is the fourth term in the expansion of ?

1) 2) 3) 4)

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Summary

Exit Ticket

The fourth term in the expansion of is

1)

2)

3)

4)

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Day 4: Algebra2/Trig Chapter 16-2: Bernoulli Binomial Probability: “EXACTLY” Probability is a very complicated subject. There is a simple situation called a “Bernoulli Trial,” named after the famous mathematician Daniel Bernoulli. A Bernoulli trial is when the only two outcomes is something that can be defined as a “success” and the opposite of that occurring is called a “failure,” and those two probabilities add up to 1.

Example 1: A fair coin is tossed 5 times. What is the probability that it lands tails up exactly 3 times?

(1) ( )1

2

3 (2) 101

2

5( ) (3) 3

5 (4) 10

1

2

3( )

Solution: n= r= p= q=

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Example 2: The probability that Kyla will score above a 90 on a mathematics test is

. What is the

probability that she will score above a 90 on three of the four tests this quarter?

(1) 4 3

3 14

5

1

5C ( ) ( ) (2)

3

4

4

5

1

5

3 1( ) ( ) (3) 4 3

1 34

5

1

5C ( ) ( ) (4)

3

4

4

5

1

5

1 3( ) ( )

Solution: n= r= p= q= Problems:

1. If the probability that the Islanders will beat the Rangers in a game is

, which expression

represents the probability that the Islanders will win exactly four out of seven games in a series against the Rangers?

(1) ( ) ( )2

5

3

5

4 3 (2) 7 4

4 32

5

2

5C ( ) ( ) (3) 5 2

2 34

7

3

7C ( ) ( ) (4) 7 4

4 32

5

3

5C ( ) ( )

2. Which fraction represents the probability of obtaining exactly eight heads in ten tosses of a fair coin?

(1) 45

1 024, (2)

90

1 024, (3)

64

1 024, (4)

180

1 024,

3. Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them

with a certain trait is

. If they have four children, what is the probability that exactly three of their

four children will have that trait?

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4. If the probability that it will rain on any given day this week is 60%, find the probability it will rain exactly 3 out of 7 days this week.

5. Jim can drive a golf ball over 220 yards 40% of the time. He regularly plays on a golf course where drives of that distance are needed on 12 holes. Determine the probability that exactly 7 of his drives will be over 220 yards.

6. If a fair coin is tossed five times, the probability of getting exactly two heads is

1) 2) 3) 4)

7. The probability that Laura wins a tennis match against Jennifer is . What is the probability that Laura wins exactly three of the next four matches she plays against Jennifer?

1) 2) 3) 4) 8. In basketball, Nicole makes 4 baskets for every 10 shots. If she takes 3 shots, what is the probability that exactly 2 of them will be baskets? 1) 0.288 3) 0.600 2) 0.432 4) 0.960

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FOR BERNOULLI EXACTLY TRIAL: the VERB you are doing. Example: Flipping ONE coin SUCCESS: What outcome is a success to you. Depends on how the problem is phrased. Example: “Getting a Head” on a coin flip FAILURE: The opposite of your success. Quite literally, in this example, “NOT getting a Head” is a failure. PROBABILITY OF SUCCESS: probability of the even happening ONE TIME. Example: P(head) = ½ # of successes: the number of times the problem is asking for a “success” to happen PROBABILITY OF FAILURE: P(success) + P(failure) = 1

SUMMARY: Exit Ticket

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Day 5 - Algebra2/Trig Bernoulli Trials: AT LEAST or AT MOST If the Bernoulli problem says “at least” or “at most,” you have to add up all of the “exactly” that represent the problem. At least: At most:

Example One: If the probability of rain is 20% on any given day this week, then what is the probability that it will rain on at least 3 days of the next 5? SOLUTION: Example 2:

On any given day, the probability that the entire Watson family eats dinner together is

. Find the

probability that, during any 7-day period, the Watsons eat dinner together at most two times.

=

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1. Team A and team B are playing in a league. They will play each other five times. If the probability

that team A wins a game is

, what is the probability that team A will win at least four of the five

games? 2. Dave is the manager of a construction supply warehouse and notes that 60% of the items

purchased are heating items, 25% are electrical items, and 15% are plumbing items. Find the probability that at least three out of the next five items purchased are heating items.

3. Tim Parker, a star baseball player, hits one home run for every ten times he is at bat. If Parker

goes to bat five times during tonight’s game, what is the probability that he will hit at most 1 home runs?

4. The probability that a planted watermelon seed will sprout is

If Peyton plants seven seeds from

a slice of watermelon, find, to the nearest ten thousandth, the probability that at least five will sprout.

5. On mornings when school is in session in January, Sara notices that her school bus is late one-

third of the time. What is the probability that during a 5-day school week in January her bus will be late at most three times?

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6. The probability of rain on the last day of July is 90%. If the probability remains constant for the

first seven days of August, what is the probability that it will rain at least six of those seven days in August?

7. East West Airlines has a good reputation for being on time. The probability that one of its flights

will be on time is .91. If Mrs. Williams flies East West for her next five flights, what is the probability that at most three of them will be on time? Round your answer to the nearest thousandth.

8. Dr. Glendon, the school physician in charge of giving sports physicals, has compiled his

information and has determined that the probability a student will be on a team is 0.39. Yesterday, Dr. Glendon examined five students chosen at random. Find, to the nearest hundredth, the probability that at least four of the five students will be on a team.

9. When Joe bowls, he can get a strike (knock down all the pins) 60% of the time. How many times

more likely is it for Joe to bowl at least three strikes out of four tries as it is for him to bowl zero strikes out of four tries? Round your answer to the nearest whole number.

10. As shown in the accompanying diagram, a circular target with a radius of 9 inches has a bull’s-eye

that has a radius of 3 inches. If five arrows randomly hit the target, what is the probability that at least four hit the bull’s-eye?

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11. Dave does not tell the truth

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4 of the time. Find the probability that he will tell the truth at most twice out of the next five times.

12. A board game has a spinner on a circle that has five equal sectors, numbered 1, 2, 3, 4, and 5, respectively. If a player has four spins, find the probability that the player spins an even number no more than two times on those four spins.

SUMMARY: If the probability of rain is 20% on any given day this week, then what is the probability that it will rain on at least 3 days of the next 5? SOLUTION:

=

algebra 2/trig: chapter 16 probability - white plains … algebra 2/trig: chapter 16 probability in...

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