The additive rule for independent events states that the probability that at least one of a number of independent events will occur is equal to the sum of their individual probabilities. For example, imagine you join a scuba diving group. As a practice, two members are randomly paired each time the group meets to go diving. Understanding the dangers of scuba diving, you are concerned about being paired with a novice diver. You find out that 35% of the group members are novice divers, 50% of the members are intermediate divers, and 15% of the members are expert divers. In probability, these events are represented as: p(novice) = 35%
p(intermediate) = 50% p(expert) = 15%
The probability of being paired with either an intermediate diver or an expert diver is determined by applying the additive rule for independent events as follows:
p(intermediate or expert) = p(intermediate) + p(expert)
Thus, there is a 65% chance (50% + 15%) that you will be paired with either an intermediate or an expert diver.
When two events are not independent, the additive rule for dependent events is used to determine the probability that at least one of a number of dependent events will occur. For example, imagine you are in charge of recruiting for a local police force. An analysis shows that the most successful officers either have a college degree or have military experience. Concerned that there will not be enough qualified people to fill the openings, you decide to survey people interested in becoming an officer to assess how many people would meet the criterion of having a college degree or having military experience. You find that 60% of those interested in becoming an officer have a college degree and 40% have military experience.
Applying the additive rule for independent events would suggest that all people who are interested in becoming officers would be qualified. However, because you are dealing with two related events (that is, some people will have both a college degree and military experience), it is necessary to subtract the joint probability of having a college degree and having military experience. Imagine that the results of your survey suggest the following
joint probabilities for each possible combination of having or not having a college degree and military experience.
Military Experience No Military Experience Total
College Degree 30% 30% 60%
No College Degree 10% 30% 40%
Total 40% 60% 100%
Where:
p(college degree, military experience) = 30% p(college degree, no military experience) = 30% p(no college degree, military experience) = 10% p(no college degree, no military experience) = 30%
Applying the additive rule for dependent events you find the following: p(college degree or military experience) =
p(college degree) + p(military experience) - p(college degree, military experience)
Thus, there is a 70% chance (60% + 40% - 30%) that a person interested in becoming an officer has either a college degree or military experience.20
Self-Test: Section IV.B.5 (answers are given on page 132)
The problems below require you to apply the additive rules of probability. Try to determine the correct answer before reading the explanation given at the end of the manual.
1. Donna is not feeling well and decides to schedule a medical examination at one of two doctors’ offices. Both offices have 20 doctors working at all times. In the past, Donna has had a number of problems with general practitioners and residents. As a result, she prefers to see internists or specialists whenever possible. Unfortunately, the policy at both offices is that a patient is treated by the first available doctor. Donna decides to call both offices to find out how many general practitioners, residents, internists, and specialists work at each office. She gets the following information.
General Practitioner Resident Internist Specialist
Office A 45% 5% 35% 15%
Office B 35% 20% 40% 5%
Given this information, at which office should Donna schedule an appointment given her preference for internists and specialists? Justify your answer by applying the additive rule for independent events.
20 Logic Note. Notice that situations in which the additive rule is used have the logical form of the alternation,
2. During a morning meeting, Sharon’s supervisor tells her that she must rent temporary office space by the end of the business day. Sharon contacts a real estate agent and tells her that she is interested in renting an office that either has ample parking space or is near public transportation. In order to narrow the search, the real estate agent tells Sharon the percentage of offices with ample parking space and the percentage of offices near public transportation in each town in the area. After hearing her options, Sharon decides that the best place to rent an office is in Milltown because 50% of the offices have ample parking space and 50% of the offices are near public transportation. On the basis of this information, Sharon determines that there is a 100% chance that the office will either have ample parking space or be near public transportation. Sharon tells the real estate agent she will rent an office in Milltown. Do you agree with Sharon’s assessment that the office is guaranteed either to have ample parking space or to be near public transportation? Justify your answer by applying the additive rule of probability for dependent events.
Section IV.B.6. Summary
This part of the unit has concentrated on ways to increase the likelihood that you will make accurate inferences in everyday life. You should go back and re-read relevant sections if you feel unsure about any of the principles outlined above. The next part of the unit will outline some of the common mistakes people make when estimating the likelihood of events.