Lesson 3-6 An Exclusive Club 297 A type of diagram borrowed from set theory is often used to illustrate sample spaces in probability. Venn diagrams use circles o portray relationships between sets of objects, in this case people. Each circle represents a set, with the objects in the set, or the number of objects in the set, written inside. If two circles overlap, the region that's common to both circles represents objects that are in both sets. We'll study Venn diagrams in much greater depth in Unit 6. In this lesson, we'll first use a Venn diagram to develop a rule for working with mutually exclusive events in probability. In 2019, there were 53 Republican senators, 45. Democratic, and 2 independents. The Venn diagram below illustrates this composition, with the orange circle representing Republicans and the blue circle representing independents. The 45 written outside of the circles represents the number of senators who are neither Republican nor independent (that is, the Democrats). Republican Independent Since a senator can't be both Republican AND independent, notice that the two circles don't overlap at all. This indicates that the conditions of being a Republican senator and being an independent senator are mutually exclusive. 5. Find the probability of a senator being Republican. 45 6. Find the probability of a senator being independent. 7. How many senators are either Republican or independent? Use this to find the probability that a senator is either Repub- lican or independent. ( goitzoo0 al vailidadong of bail neo yoy eyew ow adi odimes(T . sim nobibba tail wo estiupat 8. Describe the relationship between the probabilities found in Questions 5, 6, and 7, then use the Venn diagram to explain why that makes perfect sense when events are mutually exclusive. Addition Rule for Probability (Mutually Exclusive Events) When two events A and B are mutually exclusive, the probability that A or B will occur is P(A or B) = P(A) + P(B)