Applications of Probability

Transcription

1 Analytical Geometry Applications of Probability Applications of Probability Let's play a game. Pretend you have a pair of dice and I will pay you the sum of the dice each time you roll as long as you do not roll a 1. You can stop after the first roll or you can keep rolling to add to your money pile. Here goes...on the first round your roll a 4 and a 5. That's a total of $9. Do you want to take the money or roll again and see if you can make more? Let's roll again. The next roll you get a 5 and a 2! Add that to our previous roll and you are now up to $16! Do you want to roll again or take the money? Let's roll again! On the next roll you get 3 and 2 making our jackpot worth $21. Should we quit while we are ahead or keep going? If we roll again we could add more money to our total but if we roll a 1, we could lose it all. How do you make a decision like that? We can use probability. Essential Questions What is independent and conditional probability and how can we use them to interpret data? How can we compute probability of compound events? How do we know when two events are independent? Module Minute Probability helps us make decisions about events in our lives. It is possible to find the probability of a single event or of multiple events happening at the same time. In our dice game, we can calculate the probability of rolling 1 on a single die or on both. This information might change your decision in the game. The funny thing about probability is that it is no guarantee. It will help us predict what might happen but there is a chance of anything happening in all situations. Sometimes there are events that will affect the probability of an outcome. There are different formulas that are used to determine whether the probability of an outcome is independent or if it has other events affecting its outcome. Key Words Complement Given a set A, the complement of A, denoted A or A', is the set of elements that are not members of A. Element A member or item in a set. Independent Events Events whose outcomes do not influence each other. Intersection of Sets The set of all elements contained in all of the given sets, denoted. Outcome A possible result of an experiment. Sample Space The set of all possible outcomes from an experiment. Set A collection of numbers, geometric figures, letters, or other objects that have some characteristic in common. Subset A set in which every element is also contained in a larger set. Union of Sets The set of all elements that belong to at least one of the given two or more sets denoted υ. Venn Diagram A picture that illustrates the relationship between two or more sets.

2 A handout of these key words and definitions is also available in the sidebar. What To Expect Dice Discussion The Land of Independence Assignment Probability Quiz Are you Positive? Project Applications of Probability Test To view the standards from this unit, please download the handout from the sidebar. Sets and Set Notation Set theory contains many new words, definitions, and symbols. Before we can apply the concepts, we need to know what each symbol represents. Read the following new terms and learn them so you can apply them in our future problems. Properties of Sets and Subsets The empty set is an element of every set. Any set is its own subset. Transitive Property: A = B, if and only if, Venn Diagrams Venn diagrams can be used to represent sets and the elements that they contain. They offer a great visual for what elements may be common for multiple sets. Take a look at the following 3 examples. Venn Diagram Explanation These sets have no elements in common. These sets have some elements in common.

3 A B Set Operations: Union and Intersection As you can see from our Venn Diagrams, sometimes two sets will intersect and some of their elements will overlap. This leads us to two new terms, unions and intersection. Union The joining of two sets...elements in a union can belong to one set OR the other set. The union of Set A and B is written: A B Intersection The set of all elements that are in BOTH set A AND set B. It is written: A B

4 Check out this fun activity to practice your understanding of set theory. Dice Discussion What would you estimate the probability of two dice showing at least one odd number? Explain your reasoning. A rubric for the discussion is available in the sidebar. iframed activity from diagram for set theory/view Probability of Compound Events Probability is the likelihood of something happening. In the previous lesson we saw how venn diagrams overlap to represent elements that belong to both sets. These types of relationships represent unions and intersections where we can consider elements that belong to both sets or one or the other. We are going to use the same concept to learn the probability of compound events.

5 Event Definition Example Compound Event A compound event combines two or more events, using the word and or the word or. For example, if you flip a coin and roll a die, is is possible to get tails and an even number. Mutually Exclusive Events Mutually exclusive events have no common outcomes. For example, choosing a club or a heart from a deck of cards. A standard deck of playing cards has 52 cards that are split into 4 suits: hearts, diamonds, clubs, and spades. It is not possible to draw one card that is both a club and a diamond, therefore, those two events are mutually exclusive. For example, choosing a black card or an Ace. It is possible to have select a black Ace. Overlapping Events Overlapping events have at least one common outcome. Probability of Mutually Exclusive and Overlapping Events If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). This means find the probability of both event A and event B and add them together.

6 If A and B are overlapping events, then P(A or B) = P(A) + P(B) P(A and B). This means find the probability of both event A and event B, add them together and then subtract the probability of the overlapping events. Watch the presentation below to learn more about mutually exclusive and overlapping events. Independent and Dependent Events Two events are independent events if the occurrence of one event has no effect on the occurrence of the other.

7 Two events are dependent events if the occurrence of one event affects the occurrence of the other. Watch the presentation below to learn more about independent and dependent events. Watch the video below to see more examples of independent and dependent events. Recall using two way frequency tables in previous courses. These tables can help organize data so that probability of certain events can be determined. Click this link to check out the simulation: The Land of Independence Assignment

8 Select the "The Land of Independence Assignment" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment when finished. Probability Quiz It is now time to complete the "Probability" quiz. You will have a limited amount of time; please plan accordingly. Module Wrap Up Assignment Checklist In this module you were responsible for completing the following assignments. Review Dice Discussion The Land of Independence Assignment Probability Quiz Are you Positive? Project Applications of Probability Test Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Then, continue to the next page for your final assessment instructions. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may have to use prior knowledge from previous units in order to answer the question correctly. If you have any questions or concerns, please contact your instructor. Final Assessments Applications of Probability Test It is now time to complete the "Applications of Probability" Test. Once you have completed all selfassessments, assignments, and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly. Are You Positive? Project

9 Select the "Are You Positive Project" Handout from the sidebar. Record your answers in a separate document. Submit your completed project when finished. A rubric for this project is available in the sidebar.