Question

Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).

a. What are the 16 outcomes in S?

b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate?

c. Which outcomes are in the event that all four mortgages are of the same type?

d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?

e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events?

f. What are the union and intersection of the two events in parts (b) and (c)?

Short answer

a. The possible outcomes are:

\(S = \left\{ {FVVV,VFVV,VVFV,VVVF,FFFF,FFFV,FFVF,FVFF,VFFF,VVFF,FFVV,FVFV,VFVF,VVVV,VFFV,FVVF} \right\}\)

b. The possible outcomes are:

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

c. The possible outcomes are:

\(B = \left\{ {FFFF,VVVV} \right\}\)

d. The possible outcomes are,

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

e. The possible outcomes are,

\(B \cup C = \left\{ {FFFF,VVVV,FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

\(B \cap C = \left\{ {FFFF} \right\}\)

f. The possible outcomes are,

\(A \cup B = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF,VVVV} \right\}\)

\(A \cap B = \phi \)

Step-by-step solution

Given information

Each of a sample of four home mortgages is classified as fixed-rate (F) or variable rate (V).

List of the possible outcomes in sample space.

a.

The possible 16 outcomes in S is given as,

\(S = \left\{ {FVVV,VFVV,VVFV,VVVF,FFFF,FFFV,FFVF,FVFF,VFFF,VVFF,FFVV,FVFV,VFVF,VVVV,VFFV,FVVF} \right\}\)

List of the possible outcomes

b.

Let A represents the event that that exactly three of the selected mortgages are fixed.

The event A includes exactly 3 F’s included in the sample of four mortgages.

The outcomes of the event that exactly three of the selected mortgages are fixed is given as,

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

List of the possible outcomes

c.

Let B represents the event that all four mortgages are of the same types.

The event B includes all 4 F’s and all four V’s.

The outcomes of the event that all four mortgages are of the same types is given as,

\(B = \left\{ {FFFF,VVVV} \right\}\)

List of the possible outcomes

d.

Let C represents the event that at most one of the four is a variable-rate mortgage.

The outcomes of the event that at most one of the four is a variable-rate mortgage is given as,

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

List of the possible outcomes in union and intersection

e.

A union of two events B and C consists of all the outcomes that are either in B or C or in both events.

Referring to part c and d, the two events are,

\(B = \left\{ {FFFF,VVVV} \right\}\)

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

The union of events is represented as,

\(B \cup C = \left\{ {FFFF,VVVV,FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

The intersection of two events B and C consists of all the outcomes that are present in both events.

The intersection of the events is represented as,

\(B \cap C = \left\{ {FFFF} \right\}\)

List of the possible outcomes in union and intersection

f.

A union of two events A and B consists of all the outcomes that are either in A or C or in both events.

Referring to part b and c, the two events are,

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

\(B = \left\{ {FFFF,VVVV} \right\}\)

The union of events is represented as,

\(A \cup B = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF,VVVV} \right\}\)

The intersection of two events A and B consists of all the outcomes that are present in both events.

The intersection of the events is represented as,

\(A \cap B = \phi \)

Therefore, there are no outcomes for the intersection of the events A and B.