8.2E: Exercises  Mutually Exclusive Events and the Addition Rule
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 114^{th} 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? 
20) Are the events F, T mutually exclusive? 
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. 
Questions 25 and 26 are adapted from Introductory Statistics from OpenStax under a creative Commons Attribution 3.0 Unported License, available for download free athttp://cnx.org/content/col11562/latest u