Basic Concepts of Probability

Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.

BASIC

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.)

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.

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

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

b. No white marble is drawn.

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

a. No yellow marble is drawn.

b. The two marbles drawn have the same color.

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

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

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

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 $0.1$ .

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

b. Find $P(U)$ .

c. Find $P(V)$ .

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:

$\begin{align*} \begin{array}{l|ccccc} \text { Outcome } & m & n & q & r & s \\ \hline \text { Probablity } & 0.18 & 0.16 & 0.24 & 0.21 \end{array} \end{align*}$

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

b. Find $P(U)$ .

c. Find $P(V)$ .

APPLICATIONS

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.

a. At least one child is a girl.

b. At most one child is a girl.

c. All of the children are girls.

d. Exactly two of the children are girls.

e. The first born is a girl.

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

ADDITIONAL EXERCISES

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

Age	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.

a. The person is a smoker.

b. The person is under 30.

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

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:

Age	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.

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

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

c. The woman had no children.

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

LARGE DATA SET EXERCISES

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

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"?