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Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules Descriptionsof chancebehaviorcontaintwoparts: In Section 5.1,we usedsimulationto chance imitatebehavior.  for each outcome. that consists of two parts: a sample space A possible outcomes. The Definition: the probabilityof aparticularoutcome. Fortunately,don’twe haveto always relyon simulationsto determine Probability Models probabilitymodel sample space S is a description is adescription of some chance process of a chance process is the set of all S and and a probability

Probability Rules Probability + Give a probability model for the chance process of tworolling  Outcomes fair, six Example:Roll the Dice Sample Space 36 - sided dice – Since the dice are fair, each one that’s red and one that’s green. outcome is equallylikely. Each outcome has probability1/36.

Probability Rules Probability + Probability models us allow to find the probability of any  In the dice If and so on. Events are usually designated by capital letters, like process. That is, an event is a subset of the sample space. An Definition: Supposeevent Sinceeach outcomehas probability1/36, P( There 4 outcomesare thatresulta sum of in 5. collection of outcomes. Probability Models A event is anyevent,weits writeprobabilityas P( - is any collection of outcomes from some chance rolling example, suppose we define eventwedefine supposeexample, rolling B is defined as “sumis defined isnot5.” What isP( A A ) = ) = 4/36. ). A as “sum is5.” B)? A,B, C, P ( B ) = ) = 1 = 32/36 – 4/36

Probability Rules Probability + + Probability Tips If expressing probability as a fraction, it is acceptable to not reduce. It’s often easier to keep the denominator the same for the probability in a series of outcomes for comparison purposes...... All probability models must obey the following rules:  outcomes outcomes in common and so can never occur together. Two events are Definition: one otherone or the is occurs the sumof their individualprobabilities. If two eventshave no outcomes in the common, probability that probability eventthat the does occur. The probability that anevent does not occur is1 minus the probability that event If all the outcomes in spacesampleequally are likely,the is 1. All possible outcomes together musthaveprobabilities whose sum The probability of anyevent between is anumber 1. 0 and  Basic Rules Basic of Probability P ( A )  number of outcomes corresponding event to mutuallyexclusive (disjoint) total total number of outcomes in sample space A occurs can be found can occursbe using found the formula if they have no A

Probability Rules Probability +  • • • • •  Basic Rules Basic of Probability and Addition rule for mutuallyevents: exclusive Complementrule: In the outcomes,caselikely ofequally If For any event P S ( A is the sample space in a probability model, a space inprobability is thesample ) B  are mutually exclusive, exclusive, mutually are number of outcomes corresponding event to total total number of outcomes in sample space P ( A A , 0 ≤ or B P P ) = ( P ( S A ( ) = 1. A C P ) = 1 ) ≤ ) 1. ( A ) + – P P ( B ( A ). ) If A A

Probability Rules Probability + (b) (a) traditionalcollegeagegroup(18 23 to years). Find the probabilitythatthe chosenstudenttheis notin Showthat thisprobability isalegitimatemodel. Distance  P Eachprobability is between 0 and 1 and Probability: (yr):Agegroup (not (not 18 231 years) to = the student’s age. Here is the probability model: who is taking distance college students. Randomly select an undergraduate student Example:Distance Learning 0.57 +0.170.14 0.12 + = 1 - learning courses are rapidly gaining popularity among 18 23 to = 1 0.57 - – – learning courses for credit and record 0.57 = 0.430.57 P (18 years) to 23 24 29 to 0.17 30 39 to 0.14 40 orover 0.12

Probability Rules Probability + + Checkpoint  Choose an American adult at random. Define two events:

A= the person has a cholesterol level of 240 milligrams per deciliter of blood (mg/dl) or above (high cholesterol)

B= the person has a cholesterol level of 200 to 239 mg/dl (borderline high)

According to the American Heart Association, P(A) = 0.16 and P(B) = 0.29

1) Explain why events A and B are mutually exclusive.

A person cannot have a cholesterol level of both 240 or above and between 200 and 239 at the same time. + Checkpoint 2) Say in plain language what the event “A or B” is. What is P(A or B)?

A person has either a cholesterol level of 240 or above or they have a cholesterol level between 200 and 239. P(A or B) = P(A) + P(B) = 0.45

3) If C is the event that the person chosen has normal cholesterol (below 200 mg/dl), what’s P(C)?

P(C) = 1 – P(A or B) = 0.55