Section 4-3 Addition Rule

Key Concept This section presents the addition rule as a device for finding probabilities that can be expressed as P(A or B), the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of the procedure. The key word in this section is “or.” It is the inclusive or, which means either one or the other or both.

Compound Event Compound Event any event combining 2 or more simple events Notation P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

General Rule for a Compound Event When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find that total in such a way that no outcome is counted more than once.

Example 1: Some trees in a forest were showing signs of disease Example 1: Some trees in a forest were showing signs of disease. A random sample of 200 trees of various sizes was examined yielding the following results: Find the probability of selecting a tree that is small or disease-free? Remember that or means, small, disease-free, or both: So, there are 40 trees that are small 105 trees that are disease-free, but we counted the trees that are small and disease-free twice, so we have to subtract them off. 40 + 105 – 24 = 121

Compound Event Formal Addition Rule P(A or B) = P(A) + P(B) – P(A and B) where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.

Example 2: Some trees in a forest were showing signs of disease Example 2: Some trees in a forest were showing signs of disease. A random sample of 200 trees of various sizes was examined yielding the following results: Find the probability of selecting a tree that is medium or diseased? Use the formal addition rule:

Compound Event Intuitive Addition Rule To find P(A or B), find the sum of the number of ways event A can occur and the number of ways event B can occur, adding in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space.

Disjoint or Mutually Exclusive Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.) Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events

Example 3: Consider the procedure of randomly selecting 1 of the 200 trees included in the table below. Determine whether the following events are disjoint. A: Getting a tree that is diseased. B: Getting a medium tree. In the table above we see that there are 37 trees that are diseased and 92 medium sized trees. The even of getting a tree that is diseased and getting a medium sized tree can occur at the same time (because there are 14 trees that are diseased and medium sized). Because those events overlap, they can occur at the same time and we say that the events are not disjoint.

Example 4: Determine whether the two events are disjoint for a single trial. Hint: Consider “disjoint” to be equivalent to “separate” or “not overlapping.” A: Randomly selecting a physician at Rush Hospital in Chicago and getting a surgeon. B: Randomly selecting a physician at Rush Hospital in Chicago and getting a female. It is possible for a surgeon to be female so these two events are not disjoint.

Example 5: Determine whether the two events are disjoint for a single trial. Hint: Consider “disjoint” to be equivalent to “separate” or “not overlapping.” A: Randomly selecting a corvette from the Chevrolet assembly line and getting one that is free of defects. B: Randomly selecting a corvette from the Chevrolet assembly line and getting one with a dead battery. These two events must be disjoint because if the car is free of defects then it cannot possibly have a dead battery because then it would be considered not free of defects.

P(A) and P(A) are disjoint Complementary Events P(A) and P(A) are disjoint It is impossible for an event and its complement to occur at the same time.

Rule of Complementary Events P(A) + P(A) = 1 P(A) = 1 – P(A)

Venn Diagram for the Complement of Event A

Example 6: FBI data show 62. 4% of murders are cleared by arrests Example 6: FBI data show 62.4% of murders are cleared by arrests. We can express the probability of a murder being cleared by an arrest as P(cleared) = 0.624. For a randomly selected murder, find P(cleared).

Example 7: A Pew Research Center poll showed that 79% of Americans believe that it is morally wrong to not report all income on tax returns. What is the probability that an American does not have that belief?

There are 512 challenges that were not successful so: Example 8: Use the table below, which summarizes challenges by tennis players (based on data reported by USA Today). The results are from the first U.S. Open that used the Hawk-Eye electronic system for displaying an instant replay used to determine whether the ball is in bounds. In each case, assume that one if the challenges is randomly selected. a) If S denotes the event of selecting a successful challenge, find P(S). b) Find the probability that the selected challenge was made by a man or was successful. Was the challenge to the call successful? Yes No Men 201 288 Women 125 224 Total 326 512 Total 489 349 838 There are 512 challenges that were not successful so: Use the formal addition rule: