Probability Chapter 11

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?

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

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?

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.

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?

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?

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.

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

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?

Aim #11-2: How do we find the number of permutations of a set of objects ? Investigation See Easiteach

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

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.

Permutations Using A Diagram Is the line up (Ryan, Emily, Justin) different from the line up (Ryan, Justin, Emily)? Explain.

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 = 5, 040 different ways

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

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!= = 40, 320 different orders.

Permutations Using Factorials Simplify each expression. 1.2! 2.6! 3.4! 4.Find the number of ways you can arrange ten books on a bookshelf.

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 25 P 2.

Permutation Notation The expression n P r represents the number of permutations of n objects chosen r at a time. Example: 25 P 2 = = objects groups of 2

Using Permutation Notation Simplify 15 P 3. = = 2, 730 permutations of 15 items

Using Permutation Notation Simplify each expression.

Summary: Answer 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.

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.

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

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.

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

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 4 C 2.

Using Combination Notation The expression n C r represents the number of combinations of n objects chosen r at a time.

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. Using Combination Notation

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

Using Combination Notation Simplify each expression.

Summary: Answer in complete sentences. Explain what the combination formula means.

Aim #11-4: How do we find experimental probability?

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.

Experimental Probability P (Event) =

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.

Finding Experimental Probability Solution:  0.249= 25%

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.

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

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)=

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?

Identifying the Type of Probability Solution: The survey records actual responses from Mayville residents. It represents experimental probability.

Identifying the Type of Probability Decide whether each probability is experimental or theoretical. Explain your answers. a.A bag contains two red marbles and three white marbles. P(red) is 2/5. a.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.

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.

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)

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

Complements in Probability What is the probability of not rolling a 4 or 5 on a number cube?

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.

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 3:1.

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.

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

Summary: Answer 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.

Aim #11-5: How do we find the probability of independent and dependent events?

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.

Independent Event If A and B are independent events, then P(A, then B) = P(A) P(B)

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) =

Practice:

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.

Dependent Events If A and B are dependent events, then P(A, then B) = P(A) P(B, after A).

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)=

Probability of Dependent Events From the above example, find the probability that a boy and then a girl are selected.

Practice:

Summary: Answer in complete sentences. Contrast the formula for finding the probability of independent and dependent events.

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.

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.

Determining Random Samples Tell whether or not the following surveys are random samples. Describe the population of the samples. a.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.

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.

Determining Random Samples To find out the type of music people in a city prefer, you survey people who are years old. Is the sample random? Explain. Describe the population of the sample.

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.

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.

Methods of Sampling for Conducting Surveys

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.

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.

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.

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

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.

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

Real-World Problem Solving a.To find out what type of music people 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.

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.

Real-World Problem Solving Look at the clipboard, and determine whether each question is biased or not. Explain your answer.

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

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

Summary: Answer in complete sentences. Explain some important steps in planning a survey. Explain how to decide whether or not a question is biased.