3-1. Exercises

1. BASIC

Ex 1. A box contains 10 white and 10 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time. (To draw “with replacement” means that the first marble is put back before the second marble is drawn.)

Ex 2. A box contains 16 white and 16 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time. (To draw “with replacement” means that each marble is put back before the next marble is drawn.)

Ex 3. A box contains 8 red, 8 yellow, and 8 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time.

Ex 4. A box contains 6 red, 6 yellow, and 6 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time.

Ex 5. In the situation of Exercise 1, list the outcomes that comprise each of the following events.

1. At least one marble of each color is drawn.

2. No white marble is drawn.

Ex 6. In the situation of Exercise 2, list the outcomes that comprise each of the following events.

1. At least one marble of each color is drawn.

2. No white marble is drawn.

3. More black than white marbles are drawn.

Ex 7. In the situation of Exercise 3, list the outcomes that comprise each of the following events.

1. No yellow marble is drawn.

2. The two marbles drawn have the same color.

3. At least one marble of each color is drawn.

Ex 8. In the situation of Exercise 4, list the outcomes that comprise each of the following events.

1. No yellow marble is drawn.

2. The three marbles drawn have the same color.

3. At least one marble of each color is drawn.

Ex 9. Assuming that each outcome is equally likely, find the probability of each event in Exercise 5.

Ex 10. Assuming that each outcome is equally likely, find the probability of each event in Exercise 6.

Ex 11. Assuming that each outcome is equally likely, find the probability of each event in Exercise 7.

Ex 12. Assuming that each outcome is equally likely, find the probability of each event in Exercise 8.

Ex 13. A sample space is $S=\{a,b,c,d,e\}$ . Identify two events as $U=\{a,b,d\}$ and $V=\{b,c,d\}$ . Suppose $P(a)$ and $P(b)$ are each 0.2 and $P(c)$ and $P(d)$ are each 0.1.

1. Determine what $P(e)$ must be.

2. Find $P(U).$

3. Find $P(V).$

Ex 14. A sample space is $S=\{u,v,w,x\}$ . Identify two events as $A=\{v,w\}$ and $B=\{u,w,x\}$ . Suppose $P(u)=0.22$ , $P(w)=0.36$ , and $P(x)=0.27$ .

1. Determine what $P(v)$ must be.

2. Find $P(A).$

3. Find $P(B).$

Ex 15. A sample space is $S=\{m,n,q,r,s\}$ . Identify two events as $U=\{m,q,s\}$ and $V=\{n,q,r\}$ . The probabilities of some of the outcomes are given by the following table:

Outcome           m         n          q         r         s
Probablity      0.18     0.16                 0.24     0.21

1. Determine what $P(q)$ must be.

2. Find $P(U).$

3. Find $P(V).$

Ex 16. A sample space is $S=\{d,e,f,g,h\}$ . Identify two events as $M=\{e,f,g,h\}$ and $N=\{d,g\}$ . The probabilities of some of the outcomes are given by the following table:

Outcome          d       e      f       g       h
Probablity    0.22    0.13   0.27            0.19

1. Determine what $P(g)$ must be.

2. Find $P(M).$

3. Find $P(N).$

2. APPLICATIONS

Ex 17. The sample space that describes all three-child families according to the genders of the children with respect to birth order was constructed in Note 3.9 "Example 4". Identify the outcomes that comprise each of the following events in the experiment of selecting a three-child family at random.

1. At least one child is a girl.

2. At most one child is a girl.

3. All of the children are girls.

4. Exactly two of the children are girls.

5. The first born is a girl.

Ex 18. The sample space that describes three tosses of a coin is the same as the one constructed in Note 3.9 "Example 4" with “boy” replaced by “heads” and “girl” replaced by “tails.” Identify the outcomes that comprise each of the following events in the experiment of tossing a coin three times.

1. The coin lands heads more often than tails.

2. The coin lands heads the same number of times as it lands tails.

3. The coin lands heads at least twice.

4. The coin lands heads on the last toss.

Ex 19. Assuming that the outcomes are equally likely, find the probability of each event in Exercise 17.

Ex 20. Assuming that the outcomes are equally likely, find the probability of each event in Exercise 18.

3. ADDITIONAL EXERCISES

Ex 21. The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage:

	Tobacco Use
Age	Smoker	Non-smoker
Under 30	0.05	0.20
Over 30	0.20	0.55

A person is selected at random. Find the probability of each of the following events.

1. The person is a smoker.

2. The person is under 30.

3. The person is a smoker who is under 30.

Ex 22. The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (A, B, C, or None) and opinion on a bond issue:

	Opinion
Affiliation	Favors	Opposes	Undecided
A	0.12	0.09	0.07
B	0.16	0.12	0.14
C	0.04	0.03	0.06
None	0.08	0.06	0.03

A person is selected at random. Find the probability of each of the following events.

1. The person is affiliated with party B.

2. The person is affiliated with some party.

3. The person is in favor of the bond issue.

4. The person has no party affiliation and is undecided about the bond issue.

Ex 23. The following two-way contingency table gives the breakdown of the population of married or previously married women beyond child-bearing age in a particular locale according to age at first marriage and number of children:

	Number of Children
Age	0	1 or 2	3 or More
Under 20	0.02	0.14	0.08
20–29	0.07	0.37	0.11
30 and above	0.10	0.10	0.01

A woman is selected at random. Find the probability of each of the following events.

1. The woman was in her twenties at her first marriage.

2. The woman was 20 or older at her first marriage.

3. The woman had no children.

4. The woman was in her twenties at her first marriage and had at least three children.

Ex 24. The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to highest level of education and whether or not the individual regularly takes dietary supplements:

	Use of Supplements
Education	Takes	Does Not Take
No High School Diploma	0.04	0.06
High School Diploma	0.06	0.44
Undergraduate Degree	0.09	0.28
Graduate Degree	0.01	0.02

An adult is selected at random. Find the probability of each of the following events.

1. The person has a high school diploma and takes dietary supplements regularly.

2. The person has an undergraduate degree and takes dietary supplements regularly.

3. The person takes dietary supplements regularly.

4. The person does not take dietary supplements regularly.

4. LARGE DATA SET EXERCISES

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Note: These data sets are missing, but the questions are provided here for reference.

Ex 25. Large Data Sets 4 and 4A record the results of 500 tosses of a coin. Find the relative frequency of each outcome 1, 2, 3, 4, 5, and 6. Does the coin appear to be “balanced” or “fair”?

Ex 26. Large Data Sets 6, 6A, and 6B record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate.

1. Find the probability that a randomly selected voter among these 400 prefers Candidate A.

2. Find the probability that a randomly selected voter among the 200 who live in Region 1 prefers Candidate A (separately recorded in Large Data Set 6A).

3. Find the probability that a randomly selected voter among the 200 who live in Region 2 prefers Candidate A (separately recorded in Large Data Set 6B).