COMPOUND EVENTS. Judo Math Inc.

Transcription

1 COMPOUND EVENTS Judo Math Inc.

2 7 th grade Statistics Discipline: Black Belt Training Order of Mastery: Compound Events 1. What are compound events? 2. Using organized Lists (7SP8) 3. Using tables (7SP8) 4. Using tree diagrams (7SP8) 5. Choose your method 6. Using simulation (7SP8) Welcome to the Black Belt Last discipline you learned a lot about probability OR the likelihood that an event will occur. You did a lot of work with dice, spinners, pennies, etc to try to predict certain outcomes. We also realized that we can figure out the mathematical likelihood that something will happen (theoretical probability), but that may be different from what actually happens when you conduct the experiment (experimental likelihood). In this, your last and final belt packet, you are going to do some modeling with probability situations that are a little more challenging. These situations are called compound events and instead of asking questions about one thing going on, we are going to ask questions about two things going on like two dice, or two spinners, or even more crazy things! In seventh grade, you are just going to do a lot of modeling and experimenting with this, but as you move to 8, 9 and 10 grade you will learn a lot more of the math behind compound events (or you can google it right now and start learning more about it!) In this discipline you are going to learn 4 different methods of understanding probability of compound events and determining the sample space of the event. My favorite is the tree diagram pictured here to the right, but you will have to decide for yourself which modeling tool works best for your amazing brain! Good luck grasshopper. Standards Included: 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 7.SP.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 7.SP.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. 7.SP.8c Design and use a simulation to generate frequencies for compound events. Judo Math Inc.

3 1. What are compound events? Compound events are the combined probability of two or more events. Independent Events Dependent Events Events that are NOT affected by previous or future events. Like tossing a coin, the probability is ALWAYS ½ no matter how many times you have already tossed heads Events that are affected by previous events what s the probability that the 3 rd card will be an Ace. This is dependent on the events before it. Just as with simple events, you determine the probability of compound events by finding all of the possible outcomes and the number of ways the given event can occur. Instead of P(A), however, you will probably have something like P(A,B) which would mean Probability of A then B see, two events! Check out these scenarios and classify them as compound or simple probability problems. Also state whether they are independent or dependent events. Justify each of your solutions: You are about to attack a bad guy in a role playing game. You will throw two dice, one numbered 1 to 10 and the other with the letters A through G. What is the probability that you will roll a 6 and a D? What are the odds that you would reach into your sock drawer and pull out a pair of orange socks if you have 3 red pair, 10 white pair, and 4 orange pair? A shuffled deck of cards is placed face-down on the table. It contains 7 hearts, 3 diamonds, 6 clubs and 5 spades. What is the probability that the top two cards are one of the hearts followed by one of the spades? Elizabeth wrote a computer program that generates two random numbers between 1 and 12. When she runs it, what is the probability that the first value will be more than 3 and the second will be less than 3? 1

4 2. Using organized Lists (7SP8) One strategy for solving probability problems that involve compound events is the use of an organized list. In an organized list, you simply make a list of all of the possible outcomes. Keeping it organized, however, it key or you are likely to miss one of the outcomes! Example: Your 3 friends, Allison, Bobby, and Cam are running a race. What is the probability that they will finish in alphabetical order? P(A, B, C) 1 st : figure out all of the possibilities (the solution set?) (A, B, C) (A, C, B) (B, A, C) (B, C, A) (C, B, A) (C, A, B) What makes this list organized?! 2 nd : How many meet the criteria (alphabetical order) only 1! 3 rd : Set up the fraction as 1/6 or 17%. Now try solving the following probability problems using an organized list: 1. In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). What is the likelihood that she would randomly draw out the same color of pants and shirt? 2

5 2. You are about to attack a bad guy in a role playing game. You will throw two dice, one numbered 1 to 10 and the other with the letters A through G. What is the probability that you will roll a 6 and a D? 3. A shuffled deck of cards is placed face-down on the table. It contains 7 hearts, 3 diamonds, 6 clubs and 5 spades. What is the probability that the top two cards are one of the hearts followed by one of the spades? 3

6 4. Elizabeth wrote a computer program that generates two random numbers between 1 and 12. When she runs it, what is the probability that the first value will be more than 3 and the second will be less than 3? 5. Above we determined that this scenario is NOT a compound event: What are the odds that you would reach into your sock drawer and pull out a pair of orange socks if you have 3 red pair, 10 white pair, and 4 orange pair? What could we do to the scenario to turn it into a compound event? 4

7 3. Using tables (7SP8) Another method of solving compound probability problems is with tables. You have probably used a lot of tables in your math career usually with ratios and maybe even a little bit with algebra. Mathematicians like tables because they help them keep things ORGANIZED! And sometimes when you are dealing with a lot of information, all you need to do to start finding patterns is to keep it organized. Check out this example: When rolling two dice, what is the probability that the sum will be larger than 10? 4 P(>10)= 6 36 = 1 6 I know these problems look familiar, but we are going to try them out now with a table instead of a list. This will help us to determine which method works best for us and for each problem! 1. In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). What is the likelihood that she would randomly draw out the same color of pants and shirt? 5

8 2. You are about to attack a bad guy in a role playing game. You will throw two dice, one numbered 1 to 10 and the other with the letters A through G. What is the probability that you will roll a 6 and a D? 3. A shuffled deck of cards is placed face-down on the table. It contains 7 hearts, 3 diamonds, 6 clubs and 5 spades. What is the probability that the top two cards are one of the hearts followed by one of the spades? 6

9 4. Elizabeth wrote a computer program that generates two random numbers between 1 and 12. When she runs it, what is the probability that the first value will be more than 3 and the second will be less than 3? 7

10 4. Using tree diagrams (7SP8) An organized list can get a little confusing when you get too many options and arrangements. And a table works very well for some situations, but not very well for others. So when all else fails, hug a tree! Wait, that s not what I meant when all else fails, use a tree diagram! In this instance, there is a tool called a tree diagram that can help you! As I mentioned in the first page of this packet, that is my favorite type of diagram for solving these problems. I hope you will like it too! A fair coin is tossed 3 times, what is the possibility of getting at least one head. number of ways >1head can occur P(>1 heads)= Total number of outcomes By looking at the diagram we can see that there is at least one heads in 7 of the 8 outcomes. Therefore P(>1 heads)= In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). What is the likelihood that she would randomly draw out the same color of pants and shirt? 8

11 2. You are about to attack a bad guy in a role playing game. You will throw two dice, one numbered 1 to 10 and the other with the letters A through G. What is the probability that you will roll a 6 and a D? 3. A shuffled deck of cards is placed face-down on the table. It contains 7 hearts, 3 diamonds, 6 clubs and 5 spades. What is the probability that the top two cards are one of the hearts followed by one of the spades? 9

12 4. Elizabeth wrote a computer program that generates two random numbers between 1 and 12. When she runs it, what is the probability that the first value will be more than 3 and the second will be less than 3? 10

13 5. Choose your method (7SP8) Now that you have mastered the organized list, table, and tree diagram, we are going to work through a variety of probability problems. In each of these scenarios you can pick whichever tool works best for you to solve the problem. If you come up with a strategy that isn t any of the three we have practiced so far, AWESOME! You are becoming a true mathematician and looking for patterns around every corner Show your teacher what you have come up with. 1. How many ways can a red, blue, and green marble be pulled from a bag? 11

14 2. Rolling the Dice: 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? 3. Bag of Sweets! Joe has a bag containing 8 red sweets, 9 yellow ones and 11 green. He takes out a sweet and eats it, then, he takes out a second sweet. What is the probability that both the sweets are red? 12

15 4. The Lottery: Kent is thinking of holding a mini-lottery to raise money. Kent will sell tickets like this for $1 each. Each player must put a cross through 2 numbers on the ticket and hand it in. At the end of the week Karl will draw out two balls from a bag. Every player who has chosen the same two numbers as shown on the balls will win a cash prize of $10. (a) How many ways are there of choosing two different numbers on the ticket? Show all your work. (b) Will the lottery be a good money raiser? Describe your reasoning. 13

16 6. Charity Game: Ann is in charge of a barrel game to raise money for charities. Each barrel contains an equal number of red, green, white and black balls. The balls are buried in sawdust so that you cannot see them before you pick one out. To play the game, you give Ann your 25, then you pick one ball from each barrel. You win $5 if all three balls are the same color. (a) Calculate the probability that you will win the $5 if you play once. (b) Do you think that the barrel game will raise money for the local charities? Show your calculations (c) Ann wants to change the game so as to increase the amount of money it makes for the charities. Describe two different kinds of change that she could make to the Lucky Dip and find how much is likely to be raised for the charities after each change. Show all your calculations. 14

17 7. Computer Fight: Two teachers are fighting to use the computers. They play a dice game to determine which class will get to use the computers. If the die lands on an even number, your math teacher earns a point. If the die lands on an odd number, your English teacher, they earns a point. The first person to earn 5 points, wins the game and gets to use the computers. Currently your math teacher has 4 points and your English teacher has 2 points. What is the probability that your math teacher will win the game? What is the probability that your English teacher will win the game? 15

18 8. You Decide! Examine the tree diagram to the right. Come up with at least 3 questions that you could answer using this tree diagram. 16

19 9. Fair Game? James and Sam are playing a game with a coin and a dice labeled 1-6. They take turns tossing the coin and the number cube then they figure out the score If the coin lands on heads, the score is twice the number on the number cube. If the coin lands on tails, the score is two more than the number on the number cube. Complete this table of possible scores If the score is a prime number, James moves 2 squares on the board. If it s not, Sam moves one square on the board. What is the probability of getting a score that is a prime number? James and Sam play the game where there are 12 trials. How many squares would you expect James to move? How many squares would you expect Sam to move? Is the game fair?! 17

20 6. Using simulation (7SP8) The final lesson in probability here is simulation. In simulation, you will design and use a simulation to generate frequencies for compound events. A simulation is when you do something that represents a real life experiment without actually doing the experiment. For example, if 1/3 of the people in the city you live in have been to Mexico, you might want to ask some of them if they have been to Mexico to see how close your result is to the actual number. Instead of going and asking some people, you could take a dice and roll it once for every person you want to ask. You can say that every time you roll a 1 or 2, it represents someone who has been to Mexico, and every time you roll a 3, 4, 5, or 6, it represents someone who has not. It is easier than actually asking people, and it is a good simulation because you have the same chance of picking someone who has been to Mexico as you have of rolling a 1 or 2 on the dice. 1. Suppose each box of a popular brand of cereal contains a pen as a prize. The pens come in four colors, blue, red, green and yellow. Each color of pen is equally likely to appear in any box of cereal. Design and carry out a simulation to help you answer each of the following questions. a. What is the probability of having to buy at least five boxes of cereal to get a blue pen? What is the mean (average) number of boxes you would have to buy to get a blue pen if you repeated the process many times? b. What is the probability of having to buy at least ten boxes of cereal to get a full set of pens (all four colors)? What is the mean (average) number of boxes you would have to buy to get a full set of pens if you repeated the process many times? 18

21 2. Blood Type Simulation: Use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? 3. River Flooding: Suppose, over many years of records, a river generates a spring flood about 40% of the time. Based on these records, what is the chance that it will flood for at least three years in a row sometime during the next five years? 19

22 Probability Game Mini-Project Project Description You and a partner will be responsible for interpreting and recreating a game related to probability. Contestants should be able to play the game and you the host should be able to explain the rules, strategies, and probability of the game. Project Requirements INSTRUCTION MANUAL/POSTER o o o Instructions on how to play the game At least two mathematical models that explain the outcomes of the game An explanation of the most effective strategy for winning the game GAME BOARD/SET o Materials needed to play the game TEAM WORK o Respectfully and equally share work load 20