Question

Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S).Consider observing the direction for each of three successive vehicles.

a. List all outcomes in the event Athat all three vehicles go in the same direction.

b. List all outcomes in the event Bthat all three vehicles take different directions.

c. List all outcomes in the event Cthat exactly two of the three vehicles turn right.

d. List all outcomes in the event Dthat exactly two vehicles go in the same direction.

e. List outcomes in D’, C\( \cup \)D, and C\( \cap \)D.

Short answer

a. The outcomes of A are:

\(A = \left\{ {RRR,LLL,SSS} \right\}\)

b. The outcomes of B are:

\(B = \left\{ {RSL,RLS,LRS,LSR,SLR,SRL} \right\}\)

c. The outcomes of C are:

\(C = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR} \right\}\)

d. The outcomes of D are:

\(D = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR,LLS,LLR,RLL,SLL,LRL,SSL,SSR,SRS,SLS,LSS,RSS} \right\}\)

e. The possible outcomes are:

\(D' = \left\{ {RRR,SSS,LLL,RLS,RSL,LSR,LRS,SLR,SRL} \right\}\)

\(C \cup D = D\)

\(C \cap D = C\)

Step-by-step solution

Given information

The vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S).

List of the possible outcomes in event A

a.

Event A represents that all three vehicles go in the same direction.

The possible outcomes of event A is,

\(A = \left\{ {RRR,LLL,SSS} \right\}\)

List of the possible outcomes

b.

Event B represents that all three vehicles take different directions.

The possible outcomes of event B is,

\(B = \left\{ {RSL,RLS,LRS,LSR,SLR,SRL} \right\}\)

List of the possible outcomes

c.

Event C represents that; exactly two out of the three vehicles turn right.

The possible outcomes of event C is,

\(C = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR} \right\}\)

List of the possible outcomes

d.

Event D represents that exactly two vehicles go in the same direction.

The possible outcomes of event D is,

\(DC = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR,LLS,LLR,RLL,SLL,LRL,SSL,SSR,SRS,SLS,LSS,RSS} \right\}\)

List of the possible outcomes in different event.

e.

Referring to part c and d,

\(C = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR} \right\}\)

\(D = \left\{ {RRS,RRL,LRR,SRR,RLR,RSR,LLS,LLR,RLL,SLL,LRL,SSL,SSR,SRS,SLS,LSS,RSS} \right\}\)

The complementary of an event D consists of all the outcomes that are not contained in D.

This implies that the event\(D'\)will contain all the outcomes of the vehicles that go in the same direction or in a different direction.

\(D' = \left\{ {RRR,SSS,LLL,RLS,RSL,LSR,LRS,SLR,SRL} \right\}\)

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

\(\begin{aligned}C \cup D &= \left\{ {RRS,RRL,LRR,SRR,RLR,RSR,LLS,LLR,RLL,SLL,LRL,SSL,SSR,SRS,SLS,LSS,RSS} \right\}\\ &= D\end{aligned}\)

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

\(\begin{aligned}C \cap D &= \left\{ {RRS,RRL,LRR,SRR,RLR,RSR} \right\}\\ &= C\end{aligned}\)