EXERCISES Section 2.1 (1-10)

1

Four universities ( 1, 2, 3, and 4 ) are participating in a holiday basketball tournament.

• In the first round, 1 will play 2 and 3 will play 4.
• Then the two winners will play for the championship, and the two losers will also play.

One possible outcome can be denoted by 1324

• (1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4).

Questions:

1. List all outcomes in $\mathcal{S}$ .
2. Let $A$ denote the event that 1 wins the tournament. List outcomes in $A$ .
3. Let $B$ denote the event that 2 gets into the championship game. List outcomes in $B$ .
4. What are the outcomes in $A\cup B$ and in $A\cap B$ ? What are the outcomes in $A^{\prime }$ ?

2

Suppose that vehicles taking a particular freeway exit can

• turn right $\left(R\right)$ ,
• turn left $\left(L\right)$ ,
• or go straight $\left(S\right)$ .

Consider observing the direction for each of three successive vehicles.

1. List all outcomes in the event $A$ that all three vehicles go in the same direction.
2. List all outcomes in the event $B$ that all three vehicles take different directions.
3. List all outcomes in the event $C$ that exactly two of the three vehicles turn right.
4. List all outcomes in the event $D$ that exactly two vehicles go in the same direction.
5. List outcomes in $D^{\prime },C\cup D$ , and $C\cap D$ .

3

Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2-3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 2-3 subsystem.

The experiment consists of determining the condition of each component

• $S$ (success) for a functioning component
• $F$ (failure) for a nonfunctioning component.

1. Which outcomes are contained in the event $A$ that exactly two out of the three components function?
2. Which outcomes are contained in the event $B$ that at least two of the components function?
3. Which outcomes are contained in the event $C$ that the system functions?
4. List outcomes in $C^{\prime },A\cup C,A\cap C,B\cup C$, and $B\cap C$ .

4

Each of a sample of four home mortgages is classified as

• fixed rate $\left(F\right)$
• variable rate $\left(V\right)$ .

1. What are the 16 outcomes in $\mathcal{S}$ ?
2. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate?
3. Which outcomes are in the event that all four mortgages are of the same type?
4. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?
5. What is the union of the events in parts (c) and (d), and what is the intersection of these two events?
6. What are the union and intersection of the two events in parts (b) and (c)?

5

A family consisting of three persons (A, B, and C) goes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each member of the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member.

One outcome is $(1,2,1)$ for

• $A$ to station 1,
• $B$ to station 2,
• $C$ to station 1 .

1. List the 27 outcomes in the sample space.
2. List all outcomes in the event that all three members go to the same station.
3. List all outcomes in the event that all members go to different stations.
4. List all outcomes in the event that no one goes to station 2.

6

A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other three (3, 4,and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5 , and another is 213 .

1. List the outcomes in $\mathcal{S}$ .
2. Let $A$ denote the event that exactly one book must be examined. What outcomes are in $A$ ?
3. Let $B$ be the event that book 5 is the one selected. What outcomes are in $B$ ?
4. Let $C$ be the event that book 1 is not examined. What outcomes are in $C$ ?

7

An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate $A$ and three slips with votes for candidate $B$ . Suppose these slips are removed from the box one by one.

1. List all possible outcomes.
2. Suppose a running tally is kept as slips are removed. For what outcomes does $A$ remain ahead of $B$ throughout the tally?

8

An engineering construction firm is currently working on power plants at three different sites. Let $A\_i$ denote the event that the plant at site $i$ is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of $A\_1,A\_2$ , and $A\_3$ , draw a Venn diagram, and shade the region corresponding to each one.

1. At least one plant is completed by the contract date.
2. All plants are completed by the contract date.
3. Only the plant at site 1 is completed by the contract date.
4. Exactly one plant is completed by the contract date.
5. Either the plant at site 1 or both of the other two plants are completed by the contract date.

9

Use Venn diagrams to verify the following two relationships for any events $A$ and $B$ (these are called De Morgan’s laws):

1. ${\left(A\cup B\right)}^{\prime }=A^{\prime }\cap B^{\prime }$
2. ${\left(A\cap B\right)}^{\prime }=A^{\prime }\cup B^{\prime }$

Hint: In each part, draw a diagram corresponding to the left side and another corresponding to the right side.

10

Recall Example 2.10 first.

1. In Example 2.10, identify three events that are mutually exclusive.
2. Suppose there is no outcome common to all three of the events $A,B$, and $C$. Are these three events necessarily mutually exclusive? If your answer is yes, explain why; if your answer is no, give a counterexample using the experiment of Example 2.10.