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 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? 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.

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

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.

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).

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