Probability

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Probability

• Chapter 11

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Aim 11-1 How do we use tree diagrams and the
counting principle?

• Tree diagrams can help you figure out all the
  possibilities when you have several to choose
  from.
• Example Lunch
• Your school cafeteria offers two sandwiches,
  chicken or tuna. For drinks you have three to
  choose from milk, apple juice or water.
• So how many many ways can you choose your lunch?

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• Your school cafeteria offers two sandwiches,
  chicken or tuna. For drinks you have three to
  choose from milk, apple juice or water.
• So how many many ways can you choose your lunch?
• Use a tree diagram to show all the possibilities.
• Go to Easitouch

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• Does the order in which you list the decisions
  make a difference? Explain.
• Suppose the cafeteria offers four drinks. How
  many different lunches can you now choose?

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The Counting Principle

• Suppose there are m ways of making one choice and
  n ways of making a second choice. Then there are
  m n ways to make the first choice followed by
  the second choice.
• Example You have 6 shirts and 5 jeans. You have
  6 5 30 different outfits.

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The Counting Principle

• A greeting card software program offers 24
  different greetings, 10 different images and 8
  font styles. How many different cards can you
  make with this program?

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The Counting Principle

• A software program to design CD covers offers 240
  background, 14 font styles, and 12 song-listing
  styles. How many different covers can you make?

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The Counting Principle

• Use a tree diagram to find the number of possible
  outcomes.
• A diner offers three choices of entrees, three
  choices for the first side order and two choices
  for the second side order. Find the number of
  possible meals.

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The Counting Principle

• Which would be more useful in finding the
  probability of an event, a tree diagram or the
  counting principle? Explain.

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Summary Answer in complete sentences.

• Explain how a tree diagram shows the counting
  principle.
• You sell balloon in a different colors and a
  different shapes.
• How many different balloons do you sell?
• How many different balloons do you sell if a 10?

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Aim 11-2 How do we find the number of
permutations of a set of objects?

• Investigation
• See Easiteach

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• A permutation is an arrangement of a set of
  objects in a particular order.

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Permutations Using A Diagram

• In how many ways can Ryan, Emily and Justin line
  up in the gym class?
• Ryan, Emily and Justin can line up in six
  different ways. This means that there are six
  permutations.

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Permutations Using A Diagram

• Is the line up (Ryan, Emily, Justin) different
  from the line up (Ryan, Justin, Emily)? Explain.

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Using the Counting Principle

• At a school awards ceremony, the principal will
  present awards to seven students. How many
  different ways can the principal give out the
  awards?
• There are seven ways to give out the first award,
  six ways to give the second , and so on.
• 7 6 5 4 3 2 1 5, 040 different ways

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Using the Counting Principle

• Suppose the principal adds an award. How does
  this affect the number of different ways to give
  out the awards?

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Permutations Using Factorials

• Many CD players can vary the order in which songs
  are played. Your favorite CD has eight songs.
  Find the number of orders in which the songs can
  be played.
• 8! 8 7 6 5 4 3 2 1 40, 320 different
  orders.

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Permutations Using Factorials

• Simplify each expression.
• 2!
• 6!
• 4!
• Find the number of ways you can arrange ten books
  on a bookshelf.

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Using Permutation Notation

• A class of 25 students must choose a president
  and a vice president. There are 25 possible
  choices for the president. Then there are 24
  possible choices for the vice president. So there
  are 2524 permutations for choosing a president
  and a vice president from 25 students. You can
  write this as 25P2.

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Permutation Notation

• The expression nPr represents the number of
  permutations of n objects chosen r at a time.
• Example 25P2 25 24 600
• 25 objects groups of 2

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Using Permutation Notation

• Simplify 15P3.
• 15 14 13 2, 730 permutations of 15 items

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Using Permutation Notation

• Simplify each expression.

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SummaryAnswer in complete sentences.

• What is a permutation?
• Name at least two ways to find the number of
  permutations of a set of objects.
• Write the notation you could use to show the
  permutation of t things taken c at a time.

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Aim 11-3 How do we find combinations by using a
list?

• The pair of yogurt toppings, raisins and nuts, is
  the same as the pair of toppings, nuts and
  raisins. They form the same combination. A
  combination is a group of items in which the
  order of the items is not considered.

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Finding CombinationsThe table below contains our
yogurt toppings. How many different ways can you
choose two toppings?
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Finding Combinations

• How many different groups of three tutors can
  your teacher choose from four students?
• Make an organized list to find the number of
  combinations.

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Finding Combinations

• Why is Example 1 not solved by finding the number
  of permutations? Explain.

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Using Combination Notation

• You can also use permutations to find
  combinations.
• You can write the number of combinations of four
  yogurt toppings chosen two at a time as 4C2.

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Using Combination Notation

• The expression nCr represents the number of
  combinations of n objects chosen r at a time.

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Using Combination Notation

• A fishing boat uses 5 fishing lines. Each line
  holds one lure. There are 12 different lures. How
  many different combinations of lures can be used
  at one time?
• Find the number of ways you can choose 5 lures
  from 12.

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Using Combination Notation

• Solution
• 12C5
• If the fishing boat uses 7 lines rather than 5
  lines, are more combinations possible? Explain.

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Using Combination Notation

• Simplify each expression.

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SummaryAnswer in complete sentences.

• Explain what the combination formula means.

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Aim 11-4 How do we find experimental
probability?
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Experimental Probability

• Probability based on experimental data is called
  experimental probability.
• You find the experimental probability of an event
  by repeating an experiment, or trial, many times.
  

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Experimental Probability

• P (Event)

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Finding Experimental Probability

• The scientist Gregor Mendel crossbred green-seed
  plants and yellow-seed plants. Out of 8, 023
  crosses, 6,022plants had yellow seeds and 2,001
  had green seeds. Find the probability that a
  plant has green seeds. Write the probability as a
  percent.

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Finding Experimental Probability
Solution
?0.249 25
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Finding Experimental Probability

• Use the table at the right. What is the
  experimental probability of getting heads after
  20 tosses? Write the probability as a fraction,
  decimal and a percent.

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• You can toss a coin to find the experimental
  probability of getting heads. You can also find
  the theoretical probability without doing any
  trials because both possible outcomes (heads or
  tails) are equally likely

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• To find the theoretical probability of an event
  with equally likely outcomes, you use the formula
  you learned in Ch. 6.
• Theoretical probability P(E)

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Identifying the Type of Probability

• The table below shows the results of a survey of
  Mayville residents. Does the survey represent
  experimental or theoretical probability?

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Identifying the Type of Probability

• Solution
• The survey records actual responses from Mayville
  residents. It represents experimental probability.

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Identifying the Type of Probability

• Decide whether each probability is experimental
  or theoretical. Explain your answers.
• A bag contains two red marbles and three white
  marbles. P(red) is 2/5.
• You draw a marble out of bag, record the color,
  and replace the marble. After 8 draws, you record
  3 red marbles. P(red) is 3/8.

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Finding Complements of Odds

• The complement of an event is the opposite of
  that event.
• For example, in a coin toss, heads is the
  complement of tails. The sum of the probabilities
  of an event and its complement is 1.

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The Complement of an Event

• For an event, A and its complement, not A, P(A)
  P(not A) 1.
• To find the probability of a complement, use the
  following formula
• P(not A) 1 P (A)

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Complements in Probability

• Find the probability of not rolling a 6 with a
  number cube. Write the probability as a fraction.
• Solution
• P(not 6) 1- 5/ 6 5/6

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Complements in Probability

• What is the probability of not rolling a 4 or 5
  on a number cube?

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Odds

• Odds in favor of an event the ratio of the
  number of favorable outcomes to the number of
  unfavorable outcomes.
• Odds against an event the ratio of the number
  of unfavorable outcomes to the number of
  favorable outcomes.

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Odds

• Example
• The odds in favor of the spinner landing on red
  are 1 to 3 or 1 3. The odds against the spinner
  landing on red are 31.

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Determining Odds

• Suppose you select a ball at random from the golf
  balls shown. What are the odds in favor of
  selecting a yellow ball?
• Solution
• Since two balls are yellow and five are orange,
  the odds of selecting a yellow ball at random are
  2 5.

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Determining Odds

• What are the odds against selecting a yellow ball
  at random?

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SummaryAnswer in complete sentences.

• How does experimental probability differ from
  theoretical probability? Give an example of each.
• What is the relationship between the odds in
  favor of an event and the odds against an event?
  Give an example of each.

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Aim 11-5 How do we find the probability of
independent and dependent events?
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• Compound events are two or more related events.
• Suppose you draw a card from a stack of ten cards
  and replace it. When you draw a second card,
  there are still ten cards from which to choose.
  The compound events are independent.
• Independent events, the outcome of one event does
  not affect the outcome of a second event.

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Independent Event

• If A and B are independent events, then P(A, then
  B) P(A) P(B)

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Probability of Independent Events

• If you have 6 blue socks and 4 black socks and 10
  white socks, what is the probability of drawing a
  white sock then a white sock if you replace the
  first?
• P(white, then white)

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Practice
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Dependent Events

• Suppose you draw a card from a stack of ten cards
  and do not replace it. When you draw a second
  card, there are fewer cards from which to choose
• These compound events are dependent events. For
  Dependent events the outcome of the one event
  affects the outcome of a second event.

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Dependent Events

• If A and B are dependent events,
• then P(A, then B) P(A) P(B , after A).

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Probability of Dependent Events

• Two girls and three boys volunteer to speak at a
  school assembly. Their names are put in a hat.
  One name is selected at random and not replaced.
  Then another name I selected. Find the P(girl,
  then girl).
• P(girl) P(girl, after girl)
• P(girl, then girl)

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Probability of Dependent Events

• From the above example, find the probability that
  a boy and then a girl are selected.

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Practice
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SummaryAnswer in complete sentences.

• Contrast the formula for finding the probability
  of independent and dependent events.

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Aim 11-7 How do we plan a survey?

• Statisticians collect information about specific
  groups using a survey. Any group of objects or
  people in a survey is called a population.

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• Sometimes a population includes too many people
  to survey. So you look at a sample of the
  population. A sample is a part of the population.
• In a random sample, each object in the population
  has an equal chance of being selected.

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Determining Random Samples

• Tell whether or not the following surveys are
  random samples. Describe the population of the
  samples.
• At a game show, five people in the audience are
  selected to play based on their seat number.
• This is a random sample. The population is the
  audience.

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Determining Random Samples

• Tell whether or not the following surveys are
  random samples. Describe the population of the
  samples.
• b. A student interviews several people in his art
  class to determine the movie star most admired by
  the students at school.
• This is not a random sample. The students in the
  art class may not represent the views of all the
  students at school. The population is the
  students at school.

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Determining Random Samples

• To find out the type of music people in a city
  prefer, you survey people who are 18-30 years
  old. Is the sample random? Explain.
• Describe the population of the sample.

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• There are other ways to sample a population.
• In a systematic sample, the members of a survey
  population are selected using a system of
  selection that depends on a random number.

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• In a stratified sample, members of the survey
  population are separated into groups to ensure a
  balanced sample. Then a random sample is selected
  from each group.

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Methods of Sampling for Conducting Surveys
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Real-World Problem Solving

• You think the school bus makes stops that are too
  far apart. You want to see if the riders on all
  buses agree. Tell whether each survey plan
  describes a good sample and, if so, name the
  method of sampling used.
• a. Randomly interview 50 people walking on the
  street.

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• Solution
• This sample will probably include people who are
  not bus riders. It is not a good sample because
  it is not taken from the population you want to
  study.

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• You think the school bus makes stops that are too
  far apart. You want to see if the riders on all
  buses agree. Tell whether each survey plan
  describes a good sample and, if so, name the
  method of sampling used.
• b. Compile a list of all bus riders by grade
  level. Put each name on a slip of a paper into
  the appropriate grade-level box. Select ten names
  from each box to survey.

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• Solution
• This is a good sample. It is an example of a
  stratified sample.

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• You think the school bus makes stops that are too
  far apart. You want to see if the riders on all
  buses agree. Tell whether each survey plan
  describes a good sample and, if so, name the
  method of sampling used.
• c. Pick four buses at random. Interview every
  fifth rider boarding the bus.

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• Solution
• This is a good sample. It is an example of a
  systematic sample.

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Real-World Problem Solving

• a.To find out what type of music people 12-16
  years old prefer, you survey people at random at
  a local art museum. Is this a good sample?
  Explain your reasoning.
• b. Describe another survey plan for part(a) that
  uses systematic or stratified sampling.
• c. Which survey method is easier to conduct?
  Explain.

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Determining Biased Questions

• Unfair questions in a survey are biased
  questions. They make assumptions that may or may
  not be true.
• Biased questions can also make one answer seem
  better than another.

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Real-World Problem Solving

• Look at the clipboard, and determine whether each
  question is biased or not. Explain your answer.

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Real-World Problem Solving

• This question is unbiased. It does not try to
  persuade you one way or the other.
• This question is biased. It makes in-line skating
  rink A seem more appealing than skating rink B.

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• c. This question is biased. It assumes you either
  in-line skate or ice skate.

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SummaryAnswer in complete sentences.

• Explain some important steps in planning a
  survey.
• Explain how to decide whether or not a question
  is biased.