Question

A man starts a random walk at a lamp post.

a)What is the probability he ends up back at the lamp post after 8 steps?

b)There is a wall 3 steps down the path. What is the probability he ends up back at the lamp post after 10 steps? Within 3 steps of the lamp post after 10 steps?

c)There is a pit of alligators 3 steps down the path. If he takes 10 steps, what is the probability the man falls into the pit and is eaten?

Answer 1

a) total number of steps that need to be taken is 8. In order to reach the lamp post back again, 4 forward steps and 4 backward steps need to be taken

therefore, the number of ways he can reach lampost back again = 8!/4!*4!

Total number of ways he can take 8 steps = 2^8

==> Probability that the man comes back = (8!/4!*4!)*1/2^8

b) Now the total number of steps = 10, therefore the person has to take 5 steps forwards anf 5 steps back ward, but there is a wall 3 steps down the road. First we calculate all the possible ways to reach lamp post in 10 steps and then subtract the combinations in which he takes more than 3 steps down the road

the number of ways he can reach lamppost in 10 steps = 10!/5!*5! = 252

the number of ways he takes more than 3 steps down the road = (6!/3!*3!)*2! = 40

therefore, probability = (252-40)/2^10 = 0.21

c) Among the 10 steps, he has to take 7 steps forward and 3 steps backward to be eaten by alligators

Total number of ways in which one can take 7 steps forward and 3 steps backward = 10!/(7!*3!) = 120

But we need to subtract the cases in which he takes more than 4 steps forward at a time = (7!/4!*3!)*2! =70

therefore the probability = (120-70)/2^10 = 0.049