What Is the Probability a Driver Returns to Their Original Lane After 4 Minutes?

[Alexsandro]

Hello, I tried to model and answer this question, but I didn't get success. Could someone help me ?

Question:

A California driver decides that he must switch lanes every minute to get ahead. If he is on 4-lane divided highway and does this at random, what is the probability that he is back on his original lane after 4 minutes (assuming no collision) ? [HINT: the answer depends on whether he starts on an outside or inside lane.]

 

[AKG]

https://www.physicsforums.com/showthread.php?t=90033

 

[Mithal]

If the driver is outside at the far left , you have these possibilities

R,R,R,L
R,R,L,L
R,R,L,R
R,L,R,L
R,L,R,R

R IS MOVING RIGHT , L IS MOVING LEFT

He returns to his lane when the number of R equals the number of L

p= 2/5

When the driver on the second lane from left , we have these possibilities

L,R,L,R
L,R,R,L
L,R,R,R
R,L,L,R
R,L,R,L
R,L,R,R
R,R,L,R
R,R,L,L

WHEN #R=#L, He returns to his original position

p = 5/8

 

[Alexsandro]

How reach the right answer ?

If the driver is outside at the far left , you have these possibilities

R,R,R,L
R,R,L,L
R,R,L,R
R,L,R,L
R,L,R,R

R IS MOVING RIGHT , L IS MOVING LEFT

He returns to his lane when the number of R equals the number of L

p= 2/5

When the driver on the second lane from left , we have these possibilities

L,R,L,R
L,R,R,L
L,R,R,R
R,L,L,R
R,L,R,L
R,L,R,R
R,R,L,R
R,R,L,L

WHEN #R=#L, He returns to his original position

p = 5/8

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This question is in a book, that giving the answers.

The right answers are:
from an outside lane: 3/8;
from an inside lane: 11/16.