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Events and their probability Basic relationships of probability Conditional probability

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## Events and their probability Basic relationships of probability Conditional probability

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**Events and their probability**• Basic relationships of probability • Conditional probability**Events and theirprobabilities**• An event is a collection of sample points. • The probability of an event is equal to the sum of the probabilitiesof the sample points in the event.**If we can identify all the sample points in an event, and**assign probabilities to each, we can compute the probability of an event.**Events and Their Probabilities**Event M = Markey Oil is Profitable M = {(10, 8), (10, -2), (5, 8), (5, -2)} P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) = .20 + .08 + .16 + .26 = .70**Events and Their Probabilities**Event C = Collins Mining Profitable C = {(10, 8), (5, 8), (0, 8), (-20, 8)} P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8) = .20 + .16 + .10 + .02 = .48**Basic Relationships of Probability**We can use these rules to compute the probability of an even when we don’t know the probability of all the sample points. • Complement of an event • Union of two events • Intersection of 2 events • Mutually exclusive events**Complement of an Event**The complement of event A (denoted by AC) includes all sample points outside of event A. Complement of A Event A AC Venn diagram Sample space**Computing Probability Using the Complement**We’re trying to get you to think of probability as area or space Note that: Thus:**Union of Events**The union of events A and B is the event containing all points in A or B or both. The union of events A ad B is denoted by Event A Event B**Union of Two Events**Event M = Markley Oil Profitable Event C = Collins Mining Profitable MC = Markley Oil Profitable or Collins Mining Profitable MC = {(10, 8), (10, -2), (5, 8), (5, -2), (0, 8), (-20, 8)} P(MC) =P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) + P(0, 8) + P(-20, 8) = .20 + .08 + .16 + .26 + .10 + .02 = .82**Intersection of 2 Events**• The intersection of events A and B contains all sample points in both A and B. • The intersection of events A and B is denoted by Sample space Event A Event B Intersection of events A and B**Intersection of 2 Events**Event M = Markley Oil Profitable Event C = Collins Mining Profitable MC = Markley Oil Profitable and Collins Mining Profitable MC = {(10, 8), (5, 8)} P(MC) =P(10, 8) + P(5, 8) = .20 + .16 = .36**Addition Law**This law provides a means of calculating the probability of event A or event B or both occur.**Notice we must subtract**or else we will count that area twice and overstate the probability of A or B occurring Sample space Event A Event B Intersection of events A and B**Addition Law**Event M = Markley Oil Profitable Event C = Collins Mining Profitable MC = Markley Oil Profitable or Collins Mining Profitable We know: P(M) = .70, P(C) = .48, P(MC) = .36 Thus: P(MC) = P(M) + P(C) -P(MC) = .70 + .48 - .36 = .82 (This result is the same as that obtained earlier using the definition of the probability of an event.)**Mutually Exclusive Events**• Events A and B are said to be mutually exclusive if they have no sample points in common. • Events A and B are mutually exclusive if, when one event occurs, the other cannot occur Event A Event B Note that:**Conditional Probability**If the probability of being promoted to detective is influenced by whether you are a man or woman, then probability is conditional**Conditional Probability**• The notation P(A | B) reads “the probability of A given B. • Conditional probability is calculated by:**= Collins Mining Profitable**given Markley Oil Profitable Conditional Probability Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P(MC) = .36, P(M) = .70 Thus:**Multiplication Law**This allows us to compute the probability of the intersection of 2 events. That is, what is the probability that both A and B will occur? Or:**Multiplication Law**Event M = Markley Oil Profitable Event C = Collins Mining Profitable MC = Markley Oil Profitable and Collins Mining Profitable We know: P(M) = .70, P(C|M) = .5143 Thus: P(MC) = P(M)P(M|C) = (.70)(.5143) = .36 (This result is the same as that obtained earlier using the definition of the probability of an event.)**IndependentEvents**Events A and B are said to be independent if the probability of event A is not changed by the existence of event B, and vice-versa. That is: and:**Multiplication Law for Independent Events**The multiplication law can be used to determine if two events are independent**Multiplication Lawfor Independent Events**Event M = Markley Oil Profitable Event C = Collins Mining Profitable Are events M and C independent? DoesP(MC) = P(M)P(C) ? We know: P(MC) = .36, P(M) = .70, P(C) = .48 But: P(M)P(C) = (.70)(.48) = .34, not .36 Hence: M and C are not independent.