Mutually Exclusive, Dependent, and Independent Events for Beginning Statistics
Practice problems for these concepts can be found at:
- Probability Solved Problems for Beginning Statistics
- Probability Supplementary Problems for Beginning Statistics
Mutually Exclusive Events
Two or more events are said to be mutually exclusive if the events do not have any outcomes in common. They are events that cannot occur together. If A and B are mutually exclusive events then the joint probability of A and B equals zero, that is, P(A and B) = 0. A Venn diagram representation of two mutually exclusive events is shown in Fig. 4-5.
EXAMPLE 4.16 An experiment consists in observing the gender of two randomly selected individuals. The event, A, that both individuals are male and the event, B, that both individuals are female are mutually exclusive since if both are male, then both cannot be female and P(A and B) = 0.
EXAMPLE 4.17 Let event A be the event that an employee at a large company is a white collar worker and let B be the event that an employee is a blue collar worker. Then A and B are mutually exclusive since an employee cannot be both a blue collar worker and a white collar worker and P(A and B) = 0.
Dependent and Independent Events
If the knowledge that some event B has occurred influences the probability of the occurrence of another event A, then A and B are said to be dependent events. If knowing that event B has occurred does not affect the probability of the occurrence of event A, then A and B are said to be independent events. Two events are independent if the following equation is satisfied. Otherwise the events are dependent.
- P(A | B) = P(A) (4.7)
The event of having a criminal record and the event of not having a father in the home are dependent events. The events of being a diabetic and having a family history of diabetes are dependent events, since diabetes is an inheritable disease. The events of having 10 letters in your last name and being a sociology major are independent events. However, many times it is not obvious whether two events are dependent or independent. In such cases, formula (4.7) is used to determine whether the events are independent or not.
EXAMPLE 4.18 For the experiment of drawing one card from a standard deck of 52 cards, let A be the event that a club is selected, let B be the event that a face card (jack, queen, or king) is drawn, and let C be the event that a jack is drawn. Then A and B are independent events since P(A) = = .25 and P(A | B) = = .25. P(A | B) = = .25, since there are 12 face cards and 3 of them are clubs. The events B and C are dependent events since P(C) = = .077 and P(C | B) = = .333. P(C | B) = = .333, since there are 12 face cards and 4 of them are jacks.
EXAMPLE 4.19 Suppose one patient record is selected from the 125 represented in Table 4.4. The event that a patient has a history of heart disease, A, and the event that a patient is a smoker, B, are dependent events, since P(A) = = .12 and P(A | B) = = .22. For this group of patients, knowing that an individual is a smoker almost doubles the probability that the individual has a history of heart disease.
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