Interesting exercise of probability

[Alexsandro]

A snake with a little more than 4 meters takes a walk for the forest in the following way. After to walk 1 meter in definitive direction it decides aleatoriamente if she capsizes for the left or if she capsizes for the right or if she continues in the same direction, independently of the previous decisions. However, if at some moment it finds its proper tail, it does not recognize it, bites and dies poisoned. She determines the probability of that the snake walks at least 10 meters before dying.

 

[TenaliRaman]

Consider encoding the above movements this way,
1. whenever the snake moves right, mark it as -1
2. whenever the snake moves straight, mark it as 0
3. whenever the snake moves left, mark it as 1

Thus the movements of snake can be encoded as a stream of -1's,0's and 1's.
This sequence terminates whenever we see 4 consecutive 1's or -1's.
The required probability then becomes,
P(length of sequence >=10)
= 1 - P(length of sequence<10)
I think this should be computable, albeit tedious (which is why i am avoiding doing it right now anyways).

-- AI

 

[balakrishnan_v]

-- and all mirror reflections state 1 whre the last 2 metres can be oriented in any way
_| " state 2 where the last 2 metres should not be oriented parallel to the first one

=| state 3

We can formulate the following:

p1(n+1)=1/3.p1(n)+1/3.p2(n)+1/3.p3(n)
p2(n+1)=2/3.p1(n)+1/3.p2(n)+1/3.p3(n)
p3(n+1)=0.p1(n)+1/3.p2(n)+0.p3(n)

which is of the form
P(n+1)=T.P(n)
So we have
P(9)=T^9.P(0)..as we need 9 movs to be alive so that it definitely moves 1 more metre
where P(0)=[1,0,0]'

SO we get

P(9)=[0.2317 0.3136 0.1108]

or pr(living)=0.6560

 

[balakrishnan_v]

To be more precise it is
12913/19683