Question

Exercise 1. 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 A that all three vehicles go in the same direction.

(b) List all outcomes in the event B that all three vehicles take diffierent directions.i

(c) List all outcomes in the event C that exactly two of the three vehicles turn right.

(d) List all outcomes in the event D that exactly two vehicles go in the same direction.

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

 

Answer 1

Solution:

• Step1

Each of three successive vehicles can go in three directions, denote

R- vehicle turns right on a particular freeway exit.

L - vehicle turns left on a particular freeway exit.

S- vehicle goes straight on a particular freeway exit.

For example, event RLS would mean that the first vehicle

turned right, second vehicle turned left and third vehicle went straight.

(a): All possible outcomes of all three vehicles go in the same direction are

A = {RRR, LLL, SSS} . 

• Step2

(b): All possible outcomes of all three vehicles take dierent direction are

B = {RLS, RSL, LRS, LSR, SRL, SLR}, as we can see, the order of letters is important.

(c): All possible outcomes of exactly two of the three vehicles turn right means that

we need exactly two out of three letters to be "R"

C = {RRL, RRS, LRR, SRR, RSR, RLR}.

• Step3

(d): All possible outcomes of exactly two vehicles go in the same direction is

D =

{RRL, RRS, LRR, SRR, RSR, RLR, LLR, LLS, RLL, SLL, LSL, LRL, SSL, SSR, SRS, SLS, LSS, RSS} .

Here, every went should contain two of the same latter (any of the three) and the third

letter should be diferent (any of the remaining two) from the one we took in the first

step, this way we obtain that exactly two vehicles go in the same direction.

(e): D' is the complement of event D. The complement (opposite) of D means that all

vehicles go in the same direction or they go in dierent direction (to avoid that exactly

two vehicles go in the same direction). Therefore

D′ = {RRR, LLL, SSS, RLS, RSL, LRS, LSR, SRL, SLR} .

CUD is event C union event D. Event C is subset of event D (meaning that exactly two

vehicles turn right is "subset" of exactly two vehicles go in the same direction). This

indicates that C ∪ D = D. For the same reason we have that C ∩ D (C intersection D) is

C ∩ D = C.