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Learn how to identify events, assign probabilities, and understand the basic relationships using Venn diagrams and the addition law.
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III. Events and their Probabilities • An event is a collection of sample points. • The probability of any one event is equal to the sum of the probabilities of the sample points in the event. Follow the KP&L example in the text to see how they identify events and assign probabilities to them. Bring questions to class.
IV. Basic Relationships The study of probabilities can become quite complicated, but if you understand some of these building blocks, much of it becomes more digestible.
A. The Complement • Given an event A, the complement of A (Ac ) is the event consisting of all sample points not in A. • Example: The event “it rains today” has as its complement the event “it will not rain today.” • So P(A)=1-P(Ac)
Venn diagrams. John Venn (1834-1923), a British logician brought to mathematics something called the “Venn diagram” and we can use it to see how events relate to one another.
Neato! An illustration • The green area represents {S}, the sample space. Event A (it rains) is surrounded by Ac, the complement of A (it doesn’t rain). Notice how they do not overlap. They are mutually exclusive events. Both cannot simultaneously occur. Ac A If there is a 40% chance of rain, there must be a 60% chance of no rain.
B. Addition Law • This is used when you have two events and you want to know the probability of at least one of those occurring. Before we can get to this law we need to understand a couple of concepts: union and intersection of two events.
1. Union of two events 1. The union of two events is the event containing all sample points belonging to A or B, or both. Notation: AB The union is anything that occurs in the sum of those two circles, including the area of overlap. A B Smashing!
2. Intersection of two events The intersection of A and B is the event containing the sample points belonging to both A and B. Notation: AB The intersection of A and B is the overlapping area, C. For example, if A is the event a student is male, and B is the event that the student is a freshman, C is the event that they are both male and a freshman. A B C I say, that’s not too tough!
3. The Addition Law P(AB) = P(A) + P(B) - P(AB) It’s important to subtract the intersection (overlap) because we can’t add that area twice. For events that are mutually exclusive, there is no intersection, and the joint probability P(AB)=0. Thus the addition law boils down to: P(AB) = P(A) + P(B)
An example. Suppose a human resource manager provides you with the following information: • 30% of employees who left within 2 years leave because of salary reasons. • 20% left because of their work assignments. • 12% left because of both reasons. What is the probability that a worker will leave for reasons of either salary or work assignment?
Solution Begin by defining events. S= the event the employee leaves because of salary. W= the event the employee leaves because of work assignments. P(S)=.30 P(W)=.20 P(SW)=.12 The first two are marginal probabilities, the last is the joint probability. P(SW) = P(S) + P(W) - P(SW) = .30 + .20 -.12 =.38
