# 8.2E: Exercises - Mutually Exclusive Events and the Addition Rule

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## PROBLEM SET: MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE

Determine whether the following pair of events are mutually exclusive.

 1) A = {A person earns more than $25,000} B = {A person earns less than$20,000} 2) A card is drawn from a deck. C = {It is a King} D = {It is a heart}. 3) A die is rolled. E = {An even number shows} F = {A number greater than 3 shows} 4) Two dice are rolled. G = {The sum of dice is 8} H = {One die shows a 6} 5) Three coins are tossed. I = {Two heads come up} J = {At least one tail comes up} 6) A family has three children. K = {First born is a boy} L = {The family has children of both sexes}

Use the Addition Rule to find the following probabilities.

 7) A card is drawn from a deck. Events C and D are: C = {It is a king} D = {It is a heart} Find P(C $$\cup$$ D). 8) A die is rolled. The events E and F are: E = {An even number shows} F = {A number greater than 3 shows} Find P(E $$\cup$$ F). 9) Two dice are rolled. Events G and H are: G = {The sum of dice is 8} H ={Exactly one die shows a 6} Find P(G $$\cup$$ H). 10) Three coins are tossed. Events I and J are: I = {Two heads come up} J = {At least one tail comes up} Find P(I $$\cup$$ J). 11) At a college, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percent of students take Finite Mathematics or Statistics? 12) This quarter, there is a 50% chance that Jason will pass Accounting, a 60% chance that he will pass English, and 80% chance that he will pass at least one of these two courses. What is the probability that he will pass both Accounting and English?

Questions 13 - 20 refer to the following: The table shows the distribution of Democratic and Republican U.S by gender in the 114th Congress as of January 2015.

 MALE(M) FEMALE(F) TOTAL DEMOCRATS (D) 30 14 44 REPUBLICANS(R) 48 6 54 OTHER (T) 2 0 2 TOTALS 80 20 100

Use this table to determine the following probabilities.

 13) P(M $$\cap$$ D) 14) P(F $$\cap$$ R) 15) P(M $$\cup$$ D) 16) P(F $$\cup$$ R) 17) P(M $$\cup$$ R) 18) P(M $$\cup$$ F) 19) Are the events F, R mutually exclusive? Use probabilities to support your conclusions. 20) Are the events F, T mutually exclusive? Use probabilities to support your conclusion.

Use the Addition Rule to find the following probabilities.

 21) If P(E) = .5 , P(F) = .4 , E and F are mutually exclusive, find P(E $$\cap$$ F). 22) If P(E) = .4 , P(F) = .2 , E and F are mutually exclusive, find P(E $$\cup$$ F). 23) If P(E) = .3, P(E $$\cup$$ F) = .6 , P(E $$\cap$$ F) = .2, find P(F). 24) If P(E) = .4, P(F) = .5 , P(E $$\cup$$ F) = .7, find P(E $$\cap$$ F).
 25) In a box of assorted cookies, 36% of cookies contain chocolate and 12% of cookies contain nuts. 8% of cookies have both chocolats and nuts. Sean is allergic to chocolate and nuts. Find the probability that a cookie has chocolate chips or nuts (he can’t eat it). 26) At a college, 72% of courses have final exams and 46% of courses require research papers. 32% of courses have both a research paper and a final exam. Let F be the event that a course has a final exam and R be the event that a course requires a research paper. Find the probability that a course requires a final exam or a research paper.