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. (Enter your answers in set notation. Enter EMPTY or ∅ for the empty set.)
(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 different directions.
(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'.

a)

A={RRR,SSS,LLL}

b)

B={RSL,SLR,LRS,RLS,SRL,LSR}

c)

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

d)

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

e)

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

Step-by-step explanation:

Sample space=S= {RRR, RRL, RRS, RLR, RLL, RLS, RSR, RSL, RSS, LRR, LRL, LRS, LLR, LLL, LLS, LSR, LSL, LSS, SRR, SRL, SRS, SLR, SLL, SLS, SSR, SSL, SSS}

a)

Let A be the event that all three vehicles go in same direction. It means that the all three vehicles go to right or left or straight. Thus, event A can be represented as

A={RRR,SSS,LLL}

b)

Let B be the event that all three vehicles go in different direction. It means that the three vehicles can go to

1. Right, Straight, Left

2. Straight, Left, Right

3. Left, Right, Straight

4. Right, Left, Straight

5. Straight, Right, Left

6.  Left, Straight, Right

Thus, event B can be represented as

B={RSL,SLR,LRS,RLS,SRL,LSR}

c)

Let C be the event that exactly two of three vehicles turn right. It means that the three vehicles can go to

1. Right, Right , Left

2. Right , Left, Right

3. Left, Right, Right

4. Right, Right , Straight

5. Straight, Right,Right

6.  Right , Straight, Right

Thus, event C can be represented as

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

d)

Let D be the event that exactly two of three vehicles go in same direction. It means that the three vehicles can go to

1. Right,  Right, Left

2. Right, Right, Straight

3. Right, Left,  Right

4. Right, Left, Left

5. Right, Straight, Right

6. Right, Straight, Straight

7.  Left,  Right, Right

8. Left,  Right, Left

9. Left, Left, Right

10. Left, Left, Straight

11. Left, Straight, Left

12. Left, Straight, Straight

13. Straight, Right, Right

14. Straight, Right, Straight

15. Straight, Left, Left

16. Straight, Left, Straight

17. Straight, Straight, Right

18.Straight,Straight, Left

Thus, event D can be represented as

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

e)

D'=S-D= {RRR, RRL, RRS, RLR, RLL, RLS, RSR, RSL, RSS, LRR, LRL, LRS, LLR, LLL, LLS, LSR, LSL, LSS, SRR, SRL, SRS, SLR, SLL, SLS, SSR, SSL, SSS} -{RRL, RRS, RLR, RLL, RSR, RSS, LRR, LRL, LLR, LLS, LSL, LSS, SRR, SRS, SLL, SLS, SSR, SSL}

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