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Analytical GeometryApplications of Probability
Applications of ProbabilityLet's play a game. Pretend you have apair of dice and I will pay you the sum ofthe dice each time you roll as long as youdo not roll a 1. You can stop after the firstroll or you can keep rolling to add to yourmoney pile. Here goes...on the first roundyour roll a 4 and a 5. That's a total of $9.Do you want to take the money or rollagain and see if you can make more?Let's roll again. The next roll you get a 5and a 2! Add that to our previous roll andyou are now up to $16! Do you want toroll again or take the money? Let's rollagain! On the next roll you get 3 and 2making our jackpot worth $21. Should wequit while we are ahead or keep going? Ifwe roll again we could add more moneyto our total but if we roll a 1, we could loseit all. How do you make a decision likethat? We can use probability.
Essential QuestionsWhat 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 MinuteProbability helps us make decisions about events in our lives. It is possible to find the probabilityof a single event or of multiple events happening at the same time. In our dice game, we cancalculate the probability of rolling 1 on a single die or on both. This information might change yourdecision in the game. The funny thing about probability is that it is no guarantee. It will help uspredict 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 differentformulas that are used to determine whether the probability of an outcome is independent or if it
has other events affecting its outcome.
Key WordsComplement Given a set A, the complement of A, denoted A or A' , is the set of elements that are notmembers 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 incommon.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.
A handout of these key words and definitions is also available in the sidebar.
What To Expect
Dice DiscussionThe Land of Independence AssignmentProbability QuizAre you Positive? ProjectApplications of Probability Test
To view the standards from this unit, please download the handout from the sidebar.
Sets and Set NotationSet theory contains many new words, definitions, and symbols. Before we can apply the concepts, we need toknow what each symbol represents. Read the following new terms and learn them so you can apply them in ourfuture problems.
Properties of Sets and SubsetsThe empty set is an element of every set.Any set is its own subset.Transitive Property: A = B, if and only if,
Venn DiagramsVenn diagrams can be used to represent sets and the elements that they contain. They offer a great visual forwhat elements may be common for multiple sets. Take a look at the following 3 examples.
Venn Diagram Explanation
These setshave no
elements incommon.
These setshave someelements incommon.
A B
Set Operations: Union and IntersectionAs you can see from our Venn Diagrams, sometimes two sets will intersect and some of their elements willoverlap. 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
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 yourreasoning.
A rubric for the discussion is available in the sidebar.
iframed activity from http://www.oercommons.org/courses/venndiagramforsettheory/view
Probability of Compound EventsProbability is the likelihood of something happening. In the previous lesson we saw how venn diagrams overlapto represent elements that belong to both sets. These types of relationships represent unions and intersectionswhere we can consider elements that belong to both sets or one or the other. We are going to use the sameconcept to learn the probability of compound events.
Event Definition Example
CompoundEvent
Acompoundeventcombinestwo or moreevents, usingthe word andor the wordor.
For example, if you flip a coin and roll adie, is is possible to get tails and aneven number.
MutuallyExclusiveEvents
Mutuallyexclusiveevents haveno commonoutcomes.
For example, choosing a club or aheart from a deck of cards. A standarddeck of playing cards has 52 cards thatare split into 4 suits: hearts, diamonds,clubs, and spades. It is not possible todraw one card that is both a club and adiamond, therefore, those two eventsare mutually exclusive.
OverlappingEvents
Overlappingevents haveat least onecommonoutcome.
For example, choosing a black card oran Ace. It is possible to have select ablack Ace.
Probability of Mutually Exclusive and Overlapping EventsIf A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). This means find the probability ofboth event A and event B and add them together.
If A and B are overlapping events, then P(A or B) = P(A) + P(B) P(A and B). This means find the probabilityof both event A and event B, add them together and then subtract the probability of the overlappingevents.
Watch the presentation below to learn more about mutually exclusive and overlapping events.
Independent and Dependent EventsTwo events are independent events if the occurrence of one event has no effect on the occurrence of theother.
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 twoway frequency tables in previous courses. These tables can help organize data so thatprobability of certain events can be determined.
Click this link to check out the simulation:
http://www.geogebra.org/m/10673
The Land of Independence Assignment
Select the "The Land of Independence Assignment" Handout from the sidebar. Record your answersin 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 planaccordingly.
Module Wrap UpAssignment ChecklistIn this module you were responsible for completing the following assignments.
Dice DiscussionThe Land of Independence AssignmentProbability QuizAre you Positive? ProjectApplications of Probability Test
ReviewNow that you have completed the initial assessments for this module, review the lesson material with the practiceactivities and extra resources. Then, continue to the next page for your final assessment instructions.
Standardized Test PreparationThe following problems will allow you to apply what you have learned in this module to how you may seequestions asked on a standardized test. Please follow the directions closely. Remember that you mayhave 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 AssessmentsApplications 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 thismaterial, 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
Select the "Are You Positive Project" Handout from the sidebar. Record your answers in a separatedocument. Submit your completed project when finished.
A rubric for this project is available in the sidebar.
