homework 5 compound probability

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Concept 5 – Compound Probability

Week 22 – probability games, xavier and travis play cards.

• What is the probability that Travis will win?
• What is the probability that Xavier will win?
• Is this a fair game?  Why or why not?

Forth of July Fundraiser

• If you buy a scratch-off ticket, is it more likely that you will win a free raffle ticket or a cash prize?  Explain your answer.
• What is the probability that you will win something (either a free raffle entry or a cash prize)?
• What is the probability that you will win nothing at all?  To justify your thinking, write an expression to find the complement of winning something.

New Fruiti Tutti Chews Flavor

• One of the new advertisements states that if you reach into any bag of Fruiti Tutti Chews, you have a probability of pulling out a Killer Kiwi candy.  Another advertisement says that  of each bag is Ridiculous Raspberry.  Are the advertisements telling the truth?
• Alicia learns that when she opens a new bag of candy, she has a chance of pulling out a piece of Ridiculous Raspberry and a chance of pulling out a piece of Killer Kiwi.  Could she have a chance of pulling out a piece of Perfect Peach?
• When the company introduces the new flavor, it plans to make Perfect Peach of the candy in each bag.  If there is an equal amount of the remaining three flavors, what is the probability that the first piece you pull out of the bag will be Crazy Coconut? 

Rona's Clothes

• Make a list of all the possible combinations. Is there a better way to organize this list?
• How many possible outfits can she make?
• What is the probability that she will wear  both  a pajama pants and a cropped top shirt?
• What is the probability that she will not wear the cargos?
• If she were to only wear her hoodie because it is raining, how many possible outfit combinations can she make?

• You spin the two spinners at right and exactly one spinner lands on 4.
• At the car rental agency, you will be given either a truck or a sedan.  Each model comes in four colors: green, black, white, or tan.  If there is one vehicle of each color for each model available, what is the probability you will get a green truck?

5-34.  RANDOM NUMBER GENERATOR

Imagine a random number generator that produces numbers from 1 to 20.  In each game below, if the stated outcome happens, Player X wins.  If it does not, then Player Y wins.

Explore using the  5-34 Student eTool  (CPM) to randomly generate a number from 1 to 20.

a. In each case, what is the theoretical probability that Player X wins?  That Player Y wins?  Decide whether each game above is fair.

b. In which of the three games is Player X most likely to win?  Why?

c. In Game 1, the prime number game, if you play 40 times, how many times would you expect Player X to win?  What if you played 50 times?

d. Obtain a random number generator from your teacher and set it up to generate integers from 1 to 20.  Play the prime number game (Game 1) ten times with a partner.  Start by deciding who will be Player X and who will be Player Y.  Record who wins each time you play.

e. How did the experimental and theoretical probabilities of Player X’s winning from part (a) and part (d) compare?

5-35.  Janelle is going to babysit her nephew all day five times this summer.  She had the idea that one way to entertain him is to walk to McBurger’s for a Kids Meal for lunch each time.  The Kids Meal comes packed randomly with one of three possible action figures.  Janelle would like to know the probability that they get all three figures in five trips.

Explore using the  5-35 Student eTool  (CPM) to generate random numbers to simulate the problem below.

a. Call the action figures #1, #2, and #3.  Use the random number generator to simulate five trips to McBurger’s.  Did you get all three action figures?

b. Simulate another five trips to McBurger’s.  Did you get all three action figures this time?  Do the simulation at least 20 times (that is, 20 sets of 5 random numbers), keeping track of how many times you got all three action figures in five tries, and how many times you did not.

c. Use your results to estimate the probability of getting all three action figures in 5 trips.  Should Janelle be worried?

d. How could Janelle get an even more accurate estimation of the probability?

5-36.  Janelle’s aunt and uncle have three children, two of whom are girls.  Assuming that girl children and boy children are equally likely, Janelle thought that the chance of having two or more girls out of 3 children must be 50%.  Janelle’s brother thought the chance of having so many girls had to be less than 50%.

Explore using the  5-36 Student eTool  (CPM) to randomly generate numbers to indicate the number of girls and boys in a family with three children.

a. What do you think?  Make a conjecture about the probability of having two or three girls in a family of three siblings.

b. Do a computer simulation with the random number generator to estimate the probability of having two or three girls in a family of three siblings.  Use a 1 to represent a girl and a 0 to represent a boy and simulate a family of three children.  Do enough trials to get a good estimate. 

Additional Challenge 5-37.  Sophia and her brother are trying to create a fair game in which you roll two number cubes.  They cannot agree on the probability that the numbers on both number cubes will be even, so they decide to design a simulation.

Explore using the  5-37 Student eTool  (CPM) to simulate the problem below.

a. Make a conjecture.  What is the probability both dice are even?

b. Design a simulation with a random number generator.  How many random numbers do you need?  In what interval should the numbers be?  How many times will you do the simulation?

c. Set up and run the simulation that you designed with the random number generator and estimate the probability.  How does it compare with your conjecture from part (a)?

Assignment:  5.2.2 Homework

[expand title=”Week 20 – Compound Probability”]

5-43.  In Chapter 1, you met Chris and her older sister, Rachel, who made a system for determining which one of them washes the dishes each night.  Chris has been washing the dishes much more than she feels is her fair share, so she has come up with a new system.  She has proposed to Rachel that they get two coins, and each day she and Rachel will take a coin and flip their coins at the same time.  If the coins match, Chris washes the dishes; if they do not match, Rachel washes the dishes.  Explore using  CPM Probability eTool  (CPM).

Rachel thinks that this is a good idea and that her little sister is very silly!  She thinks to herself,  “Since there are two ways to match the coins, Heads- and -Heads or Tails- and -Tails, and only one non-match, Heads- and -Tails, then Chris will STILL wash the dishes more often.  Ha!”

a. Do you agree with Rachel?  Why or why not?

b. Does it matter if they flip the coins at the same time?  That is, does the result of one coin flip depend on the other coin flip?

c. What are all of the possible outcomes when the girls flip their coins?  Organize the possibilities.  Use the word “and” when you are talking about  both  one thing  and  another occurring.

d. Look at your list from part (c).  Imagine that the coins are a penny and a nickel instead of two of the same coin.  Does your list include both the possibilities of getting a heads on the penny and tails on the nickel and vice versa?  If not, be sure to add them to your list.

e. Is Rachel right?  Does this method give her an advantage, or is this a fair game?  What is the theoretical probability for each girl’s washing the dishes?

5-44.  ROCK-PAPER-SCISSORS

Read the rules for the Rock-Paper-Scissors game below.  Is this a fair game?  Discuss this question with your team.

• At the same time as your partner, shake your fist three times and then display either a closed fist for “rock,” a flat hand for “paper,” or a partly closed fist with two extended fingers for “scissors.”
• Rock beats scissors (because rock blunts scissors), scissors beats paper (because scissors cut paper), and paper beats rock (because paper can wrap up a rock). If you both show the same symbol, repeat the round.

a. While both players are making their choice at the same time, this game has  two events  in every turn.  What are the two events?

b. If you and a partner are playing this game and you both “go” at the same time, does your choice affect your partner’s choice?  Explain.

c. Are the two events in this game  dependent  (where the outcome of one event affects the outcome of the other event) or  independent  (where the outcome of one event does  not  affect the outcome of the other event)?  Explain your reasoning.

d. Work with your team to determine all of the possible outcomes of a game of rock-paper-scissors, played by two people (call them Person A and Person B).  Be sure to include the word “and.”  For each outcome, indicate which player wins or if there is a tie.  Be prepared to share your strategies for finding the outcomes with the class.

5-45. Is Rock-Paper-Scissors a fair game?  How can you tell?

5-46.  Imagine that two people, Player A and Player B, were to play rock-paper-scissors 12 times.

a. How many times would you expect Player A to win?  Player B to win?

b. Now play rock-paper-scissors 12 times with a partner.  Record how many times each player wins and how many times the game results in a tie.

c. How does the experimental probability for the 12 games that you played compare to the theoretical probability that each of you will win?  Do you expect them to be the same or different?  Why?

5-47.  Identify the situations below as either dependent or independent events.

a. Flipping a “heads” on a quarter  and  then flipping another “heads.”

b. Choosing a jack from a standard deck of cards, not putting it back in the deck,  and  then choosing a king.

c. Picking a blue marble from a bag of marbles, putting it back,  and  then picking a blue marble again.

d. Rolling a 6 on a number cube three times in a row.

5-48.  LEARNING LOG

In your Learning Log, make an entry that summarizes your understanding of independent and dependent events.  Explain how to decide if two events are independent or dependent when looking at their likelihood.  Give a few examples to support your thinking.  Title your notes “Independent and Dependent Events” and include today’s date.

Assignment:  5.2.3 Homework

Probability tables:  a new strategy for organizing all of the possibilities in a complicated game.

5-54.  TEN Os

In this game, you will create a strategy to play a board game based on your predictions of likely outcomes.  You will place ten O’s on a number line.  Then your teacher will roll two number cubes  and add the resulting numbers.  As your teacher rolls the number cubes and calls out each sum, you will cross out an O over the number called.  The goal of the game is to be the first person to cross out all ten of your O’s. Create number cubes using the  5-54 Ten 0’s Game  (CPM).

Talk with your team about the possible outcomes of this game.  Then draw a number line like the one below on your own paper.  Place a total of ten O’s on your number line.  Each O should be placed above a number.  You should distribute them based on what results you think your teacher will get.  More than one O can be placed above a number.

Follow your teacher’s instructions to play the game.

Gerald began by trying to create a table to list all of the possible combinations of rolls.  He made the table at right.

Did he list them all?  If so, how can you be sure that they are all there?  If not, give examples of a few that he has missed.

5-56.  Gerald decided that this method was taking too long, that it was too confusing, and that he made too many mistakes.  Even if he listed all of the combinations correctly, he still had to find the sums and then find the theoretical probabilities for each one.  Inspired by multiplication tables, he decided to try to make sense of the problem by organizing the possibilities in a  probability table  like the one shown below right.

b. How many total possible number combinations are there for rolling the two cubes?  Is each combination listed equally likely?  That is, is the probability of getting two 1’s the same as that of getting two 2’s or a 3 and a 1?

c. How many ways are there to get each sum?  Are there any numbers on the game board that are not possible to achieve?

d. What is the theoretical probability for getting each sum listed on the Ten O’s game board?

e. Now work with your team to determine a better strategy for Gerald to place his ten O’s on the game board that you think will help him to win this game.  Explain your strategy and your reasoning.

5-57.  Gloria and Jenny each have only one O left on their game board.  Gloria’s O is at 6, and Jenny’s is at 8.  Which student is more likely to win on the next roll?  Explain.

5-58.  Now go back and analyze the game of rock‑paper-scissors using a probability table to determine the possible outcomes.

a. Make a probability table and use it to find the probability of Player A’s winning and the probability of Player B’s winning.  Did you get the same answers as before?

b. Do the probabilities for Player A’s winning and Player B’s winning add up to 1 (or 100%)?  If not, why not?

5-59.  Imagine that you have a bag with a red block, a blue block, a green block, and a yellow block inside.  You plan to make two draws from the bag, replacing the block after each draw.

• Are these two events (the two draws of a block) independent or dependent?  Does it matter if you replace the block each time?  Why or why not?
• Find the probability of getting a red block and a blue block.  (Either color can come first.)  Be ready to share your method of organizing the possible outcomes.

Assignment:  5.2.4 Homework

[expand title=”Week 21 – Probability Trees”]

A giant wheel is divided into 5 equal sections labeled –2, –1, 0, 1, and 3.  At the Double Spin, players spin the wheel shown at right two times.   The sum of their spins determines whether they win.  Explore using  5-65 Spinner eTool  (CPM).

Work with your team to determine probabilities of different outcomes by answering the questions below.

Make a list of the possible sums you could get.

a. Which sum do you think will be the most probable?

b. Create a probability table that shows all possible outcomes for the two spins.

c. If Tabitha could choose the winning sum for the Double Spin game, what sum would you advise her to choose?  What is the probability of her getting that sum with two spins?

5-66.  Scott’s job at Crazy Creations Ice Cream Shop is to design new ice cream flavors.  The company has just received some new ingredients and Scott wants to be sure to try all of the possible combinations.  He needs to choose one item from each category to create the new flavor.

a. Without talking with your teammates, list three different combinations Scott could try.  Make sure you use the word “and.”  Then share your combinations with your study team.  How many different combinations did you find?  Do you think you found all of the possibilities?

• How many different flavor combinations are possible?  Where do you look on the diagram to count the number of complete combinations?
• Use your probability tree to help you find the probability that Scott’s final combination will include plum swirl.
• What is the probability that his final combination will include hazelnuts

5-67.  Scott’s sister loves hazelnuts and Scott’s little brother loves grape.

• Recall that events are favorable outcomes.  List all of the outcomes in Scott’s sister’s event.  List all the outcomes in Scott’s little brother’s event.
• Two events are  mutually exclusive  if they have no outcomes in common.  Do Scott’s sister and little brother have mutually exclusive events?
• What would two mutually exclusive events in the Crazy Creations Ice-Cream Shop be?

5-68.  In a power outage, Rona has to reach into her closet in the dark to get dressed.  She is going to find one shirt and one pair of pants.  She has three different pairs of pants hanging there: one black, one brown, and one plaid.  She also has two different shirts: one white and one polka dot.

• Draw a probability tree to organize the different outfit combinations Rona might choose.
• What is the probability that she will wear  both  a polka dot shirt  and  plaid pants?
• What is the probability that she will not wear the black pants?
• For what kinds of problems can you also make a probability table?  If it is possible, make a probability table for Rona’s outfits.  Which way of representing the outcomes do you like better?
• Are the events polka dot and plaid mutually exclusive?  Explain.  .
• Are the events polka dot and white mutually exclusive?  Explain.

5-69.  Represent all of the possible outcomes using a list, probability table, or probability tree.  Then find the indicated probability in each situation below.

Assignment: 5.2.5 Homework

5-76.  Nina is buying a new pet fish.  At the pet store, the fish tank has an equal number of two kinds of fish: tetras and guppies.  Each kind of fish comes in four different colors: yellow, orange, blue, and silver.  There are an equal number of each color of fish in the tank.

a. If Nina scoops out a fish at random, what is the probability that she will scoop out a silver tetra?  Show how you decided.

d. What is the area of the complete large rectangle?

5-77.  TESTING THE AREA MODEL

a. Use a systematic list or a probability tree to organize the possible color combinations she could draw.  How many are there?  What is P(blue and blue)?

b. This time, Nina made the table at right.  Based on the table, what is the probability of drawing two blue blocks?  Is this the same probability you found in part (a)?

c. Looking at her work, Edwin said,  “I think I can simplify this diagram.”   His rectangle is shown at right.  What is the area of the section representing blue and blue?  Does this match the probability that Nina found?

5-78.  The pet store sells a lot of pet food.  On a slow day at the pet store, three people buy cat food, two people buy dog food, and one person buys food for a pet snake.  If half of the customers pay with cash and half pay with credit card, what is the probability that a customer buying pet food will buy dog food with cash?  Set up an area model like Edwin’s in part (c) of problem 5‑77 to help you find the probability.

5-79.  SPINNING ODDS AND EVENS — PART 1

Your team is going to play against your teacher in a game with two hidden spinners.  Spinner A has the numbers 2, 3, and 4 on it.  Spinner B has the numbers 6, 7, and 8 on it.

The rules are:

• Spin each spinner.
• Add the results.
• If the sum is even, one team gets a point.  If the sum is odd, the other team gets a point.
• The first team to earn 10 points wins.

a. Should you choose the odd or even numbers in order to win?  Discuss the choices with your team and decide which side to take.  Be prepared to justify your choice with mathematics.

b. Play the game at least three times with your teacher.  Your teacher will spin the spinners and announce the results.  Record the results of each spin and their sum.  Is the result odd or even most often?  Does this match with your prediction?

c. Make a probability table and determine the theoretical probability for this game.

5-80.  SPINNING ODDS AND EVENS — PART 2

Now that you have played the game several times, obtain a  Lesson 5.2.6 Resource Page  from your teacher and take a close look at the hidden spinners.

a. Are the spinners different from what you expected? How? Be as specific as you can. Do you still think you made the correct choice of odd or even numbers?

b. What assumption about the spinners did you make in part (c) of problem 5‑79?

c. What is the probability of spinning each outcome on Spinner A?  On Spinner B?

a. Create a new rectangle.  Label the top row and left column with the numbers on each spinner and their probabilities.

b. Write a multiplication problem to show the probability of spinning a 3 and a 7.  Calculate P(3 and 7).

c. Complete the table to show each possible sum and its probability.

d. What is the probability of spinning an odd sum?  What is the probability of spinning an even sum?

e. Did you make the right choice of an odd or even number in problem 5-80?  Explain your reasoning.

5-82.  Additional Challenge:  Elliot loves music, especially listening to his music player on shuffle.  He has songs stored in four categories: country, blues, rock, and classical.  Two fifths of his songs are country songs, one sixth of his songs are classical, one third are blues, and the rest are rock.

a. What is the likelihood that the first two songs will be   a country song  and  then a classical?

b. What is the likelihood that  either  a country song  or  a classical song will come up first?

Assignment:  5.2.6 Homework

Compound Probability

Compound probability is the probability of two or more independent events occurring together. Compound probability can be calculated for two types of compound events, namely, mutually exclusive and mutually inclusive compound events. The formulas to calculate the compound probability for both types of events are different.

Compound probability is a concept that is widely used in the finance industry to assess risks and assign premiums to various policies. In this article, we will learn more about compound probability, its formulas, how to determine it as well as see various associated examples.

What is Compound Probability?

The compound probability of compound events (mutually inclusive or mutually exclusive) can be defined as the likelihood of occurrence of two or more independent events together. An independent event is one whose outcome is not affected by the outcome of other events. A mutually inclusive event is a situation where one event cannot occur with the other while a mutually exclusive event is when both events cannot take place at the same time. The compound probability will always lie between 0 and 1.

Compound Probability Formulas

There are two formulas to calculate the compound probability depending on the type of events that occur. In general, to find the compound probability, the probability of the first event is multiplied by the probability of the second event and so on. The compound probability formulas are given below:

Mutually Exclusive Events Compound Probability

• P(A or B) = P(A) + P(B)

Using set theory this formula is given as,

• P(A ∪ B) = P(A) + P(B)

Mutually Inclusive Events Compound Probability

• P(A or B) = P(A) + P(B) - P(A and B)
• P(A ∪ B) = P(A) + P(B) - P(A ⋂ B)

where A and B are two independent events, and P(A and B) = P(A) x P(B)

Compound Probability Example

Suppose a coin is tossed. The outcome of getting heads will be a simple event with a probability of 1 / 2. However, if the coin is tossed twice then the outcome of getting two heads will be a compound event. The compound probability of this event can be calculated as (1 / 2) x (1 / 2) = 1 / 4 or 0.25. This is an example of compound probability.

How to Find Compound Probability?

The steps to apply the compound formulas can be understood with the help of an example. Suppose the probability of Ryan failing an exam is 0.3 and the probability of Berta failing is 0.2. Then to find the compound probability of Ryan or Berta failing, the steps are as follows:

• Determine if the compound event is mutually exclusive or inclusive. This is an example of a mutually inclusive event.
• List the given probabilities. P(R) = 0.3 and P(B) = 0.2.
• Determine the correct compound probability formula. This is P(A or B) = P(A) + P(B) - P(A and B) for the given example
• Find P(A and B) which is given by P(A) x P(B). Thus, 0.2 x 0.3 = 0.6.
• Plug the values into the formula to get the result. Thus, the compound probability for the example is 0.44

Related Articles:

• Probability Rules
• Events in Probability
• Exhaustive Events
• Dependent Events
• Mathematical Induction

Important Notes on Compound Probability

• Compound probability is the likelihood of occurrence of two independent compound events together.
• Compound probability can be calculated for mutually exclusive and mutually inclusive compound events.
• P(A or B) = P(A) + P(B) and P(A or B) = P(A) + P(B) - P(A and B) are the compound probability formulas.

Examples on Compound Probability

Example 1: There are 40 girls and 30 boys in a class. 10 girls and 20 boys like tennis while the rest like swimming. If a student is selected at random then what is the probability that it will be a boy or a girl.

Solution: If a student is selected it can only be a girl or a boy. Thus, the probability that the selected student will be a girl or a boy is 1.

Answer: P(Boy or Girl) = 1

Example 2: If a dice is rolled then find the compound probability that either a 2 or 3 will be obtained.

Solution: P(2) = 1 / 6

P(3) = 1 / 6

As this is an example of a mutually exclusive event thus, the compound probability formula used is

P(2 or 3) = (1 / 6) + (1 / 6)

Answer: P(2 or 3) = 1 / 3

Example 3: Find the compound probability of selecting 5 or a black card from a deck.

Solution: Total number of cards in a deck = 52

Number of cards in each suit = 13

P(Black card) = 26 / 52

Total number of 5s in a deck = 4

P(5) = 4 / 52

Number of 5s in black cards = 2

P(Black card and 5) = 2 / 52

This is a mutually inclusive event thus, the compound probability formula is

P(Black card or 5) = P(Black card) + P(5) - P(Black card and 5)

Answer: P(Black card or 5) = 7 / 13

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FAQs on Compound Probability

What is the meaning of compound probability.

Compound probability, in probability and statistics , is the probability that describes the chance that two or more independent events will occur together. It is determined by multiplying the probabilities of the occurring events.

What is the Formula for Compound Probability?

There are two formulas available for calculating the compound probability. These are given as follows:

What is the Formula for Compound Probability in Set Theory?

The compound theory formulas expressed using set operations are given as follows:

What are the Types of Events Used in Compound Probability?

There are two types of events used in compound probability. These are mutually exclusive and mutually inclusive compound events.

Can Compound Probability be Greater Than 1?

The compound probability value will always lie between 0 and 1. 0 indicates that the event will never occur and 1 denotes that the event will definitely take place.

What is the Difference between Simple and Compound Probability?

Simple probability is used to give the likelihood that one event will take place. On the other hand, compound probability gives the probability of occurrence of more than one separate event.

How to Calculate Compound Probability?

The steps to calculate the compound probability are as follows:

• Determine if the event is mutually inclusive or mutually exclusive.
• Apply the corresponding formula.

Math Worksheets Land

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Probabilities of Compound Events Worksheets

How to Determine the Probabilities of Compound Events - A compound event is where there is a possibility of more than one possible outcome. Determining the probability of a compound event involves finding the sum of all the probabilities of the event and removing the ones that overlap. Probability is defined as the likelihood of an event to occur. It is normally written in the form of a fraction where the number of favorable outcomes is on the numerator, and the total number of outcomes is in the denominator. Probability is used for determining many things from the likelihood that someone will win a lottery or a baby will be born with a medical problem. Example of compound events are: Chances of rolling a six in a dice: 1/6 Probability of picking a heart from a deck of cards: There is a total of 52 cards out of which 13 will be heart so that probability would be 13/52. Simplifying it further, the probability will be 1/4. These worksheets and lessons will help students predict the possible outcomes of events that have multiple end results.

Aligned Standard: Grade 7 Statistics - 7.SP.C.8

• Swimming Places Step-by-step Lesson - Amanda, Bryan, and Carl are competing in a swimming event. What could possibly be their places?
• Guided Lesson - Two boys race each other. We flip a coin and rearrange the letter of a number word.
• Guided Lesson Explanation - This one gets progressively more difficult.
• Practice Worksheet - There are so many probabilities in this one that it is amazing.
• Matching Worksheet - Match the data output to the scenarios that generate them.
• Single and Compound Events Five Worksheet Pack - These questions are purposely two sentences or less to make them easy to outline.
• Problems Involving AND & OR Five Worksheet Pack - Break down all of these problems to make them easy to see in groups.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Homework 1 are things I always think about during the Olympics. Homework 2 are things I think about all the time.

• Homework 1 - Jacob, Michael, and Carl are about to compete in 100m mixed running event. These are the only three racers competing in this event. They are all working hard to win first place. How many different possible ways could the 3 runners place?
• Homework 2 - How many different ways you can arrange the letters in word "BEST"?

Practice Worksheets

In the first one, we talk about number arrangements. The second page focuses more on items of chance.

• Practice 1 - How many ways can you arrange these words?
• Practice 2 - There are 3 marbles (yellow, green, and blue). You pick all 3 marbles out of a bag in a row. How many outcomes are possible?

Math Skill Quizzes

It is very important that students read these questions carefully.

• Quiz 1 - Gabriel writes his name in the Urdu and English language with a pen and pencil. How many ways can he write his name?
• Quiz 2 - Denny chose two cards randomly from a deck. What is the probability of getting a King and a Jack without replacement?
• Quiz 3 - A dice is rolled 5 times. What is the chance of landing on a 4 all times?

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Easy Peasy All-in-One Homeschool

A complete, free online christian homeschool curriculum for your family and mine, compound probability.

In this lesson you will learn about compound events, composed of independent and dependent events. You will see how to create probability models that show the sample space for compound events and how to calculate arrangements by using factorials. You will also learn how to set up simulations for experiments and determine outcomes from these simulations.

Think about compound sentences. They are composed of at least two complete sentences. Well, a  compound event  is composed of two or more separate events. In order to find the probability of a compound event, it is important to understand if it is an  independent  or  dependent  event.

Independent events  are events in which the outcome of one event does not affect the probability of the other.

• Example : Toss a coin twice – The result of one toss has no effect on the result of the other toss.
• Example : Draw a marble from a bag of assorted marbles, replace the marble, and draw another.

Dependent events  are events in which the outcome of one event does affect the probability of the other.

• Example : Draw a marble from a bag of assorted marbles, and then draw another marble from the bag without replacing the first marble.
• Example : One student selects a book from the class library, and then another student selects a book from the remaining books.

The probability of a compound event is the ratio of favorable outcomes to total outcomes in the sample space for which the compound events happen. Remember that you can use sample spaces (tree diagrams, tables, and organized lists) to find the probability of compound events. You may also use the rules below:

• Probability of independent events (Multiply the probabilities): P (A and B) = P (A) * P (B)
• Probability of dependent events: (Calculate the probability of the first event. Calculate the probability that the second event would occur if the first event had already occurred.) In other words, multiply the probability of the first by the probability of the second AFTER the first: P(A and B) = P(A) x P(B after A)

Kelly and her friends are playing a board game. To move her game piece, she must roll the same number on two number cubes (1-6). Represent the sample space and find all of the ways that Kelly could roll the same number.

You can create a Tree Diagram to show the sample space:

Notice that there are 36 outcomes in the sample space and that there are 6 outcomes where the number cubes are the same. Kelly could roll the same number 1 out of 6 times.

You can make a table to show the sample space:

The sample space shows there are 36 outcomes and that 6 of those are doubles.

You can also use the formula for probability of independent events to find the probability. Remember that rolling two number cubes represents two independent events.

The letters in the word  mathematics  are placed in a container.

If two letters are chosen at random, what is the probability that both will be consonants?

Since the first letter drawn is not replaced, the sample space is different for the second letter, so the events are dependent. Find the probability that the first letter selected is a consonant.

The sample space has 11 outcomes. 7 of those are consonants.

If the first letter selected was a consonant, which changes the number of letters in the container to 10 and the total number of consonants to 6. Find the probability that the second letter selected is a consonant.

Now use the formula to find the probability of choosing two consonants.

Before using a strategy to find the probability of compound events, determine whether the event is dependent or independent.

Can you tell whether the following examples are independent or dependent?

• Drawing a green ball from a bucket and then drawing a yellow ball without replacing the first. 
• Spinning a 4 on a spinner three times in a row.
• Flipping a coin and getting heads-up, then flipping the same coin and getting tails-up.
• Selecting a two green marbles from a bag of assorted color marbles.  ( Check answers )

Compound Probability Practice

Two dice are rolled. what is the probability of the following outcomes.

• The sum of the number cubes is 6.
• The sum of the number cubes is at least 8.
• The sum of the number cubes is not a prime number.
• The sum of the number cubes is less than 4.
• The sum of the number cubes is a prime number.
• The sum of the number cubes is 7.
• The sum of the number cubes is less than 2.
• The sum of the number cubes is 12.

Module 5: Probabilities of Compound Events

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Module Overview

Probability is widely used in many career fields to help researchers and companies make reasonable predictions based on data collected.  Thus, it is important to understand how to calculate and evaluate probabilities for simple and compound events. 

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Unit 8 – Probability

Probability Terminology

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Compound Events

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Sums of Dice

Simulating Compound Events

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Curriculum  /  Math  /  7th Grade  /  Unit 8: Probability  /  Lesson 8

Probability

Lesson 8 of 9

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Determine the probability of compound events.

Common Core Standards

Core standards.

The core standards covered in this lesson

Statistics and Probability

7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Determine the sample space for a compound event and use lists, tables, or tree diagrams as needed to organize the sample space.

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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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A fair six-sided die is rolled twice. What is the theoretical probability that the first number that comes up is greater than or equal to the second number?

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Rolling Twice , accessed on June 14, 2017, 2:49 p.m., is licensed by Illustrative Mathematics under either the  CC BY 4.0  or  CC BY-NC-SA 4.0 . For further information, contact Illustrative Mathematics .

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the theoretical probability that Angie and Carlos are seated opposite each other?

Sitting Across from Each Other , accessed on June 14, 2017, 2:40 p.m., is licensed by Illustrative Mathematics under either the  CC BY 4.0  or  CC BY-NC-SA 4.0 . For further information, contact Illustrative Mathematics .

A standard deck of playing cards includes 52 cards, equally divided among 4 suits: hearts, diamonds, clubs, and spades.

You have two standard decks of cards. You draw a card from the first deck, record the suit, and then draw a card from the second deck and record the suit.

a.   What is the probability that you draw a heart and a diamond, in that order?

b.   What is the probability that you draw a heart and a diamond, in any order?

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Lin plays a game that involves a standard number cube and a spinner with four equal sections numbered 1 through 4. If both the cube and spin result in the same number, Lin gets another turn. Otherwise, play continues with the next player. What is the probability that Lin gets another turn?

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Topic A: Probability Models of Simple Events

Understand the probability of an event happening is a number between 0 and 1, ranging from impossible to certain.

Define probability and sample space. Estimate probabilities from experimental data.

7.SP.C.6 7.SP.C.7

Determine the probability of events.

7.SP.C.7.A 7.SP.C.7.B

Use probability to predict long-run frequencies.

Design and conduct simulations to model real-world situations.

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Topic B: Probability Models of Compound Events

Conduct simulations with multiple events to determine probabilities.

7.SP.C.8 7.SP.C.8.C

List the sample space for compound events using organized lists, tables, or tree diagrams.

Design and conduct simulations to model real-world situations for compound events.

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Chapter Review

3.1 terminology.

In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

3.2 Independent and Mutually Exclusive Events

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent

In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

3.3 Two Basic Rules of Probability

The multiplication rule and the addition rule are used for computing the probability of A and B , as well as the probability of A or B for two given events A , B defined on the sample space. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common.

3.4 Contingency Tables

There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables, also known as two-way tables, help display data and are particularly useful when calculating probabilites that have multiple dependent variables.

3.5 Tree and Venn Diagrams

A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities.

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Compound Probability Task Cards

Students will practice solving problems related to compound probability (including both independent and dependent events) with this set of 20 task cards.

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