Thread: Help me, please

Would you help me with this problem?

3 out of 4 trucks are followed by a car, while only 1 out of 5 cars are followed by a truck.
What fractions of vehicles on the road are trucks?

Thank you in advance!

Originally Posted by maria_stoeva

Would you help me with this problem?

3 out of 4 trucks are followed by a car, while only 1 out of 5 cars are followed by a truck.
What fractions of vehicles on the road are trucks?

Thank you in advance!

Pr(T | C) = 3/4.
Pr(C | T) = 1/5.

Using a Markov chain and letting n --> oo I get 4/9.

Originally Posted by mr fantastic

Pr(T | C) = 3/4.
Pr(C | T) = 1/5.

Using a Markov chain and letting n --> oo I get 4/9.

What do these probabilities represent? The probability that you have a truck given that it is followed by a car?

Originally Posted by Sampras

What do these probabilities represent? The probability that you have a truck given that it is followed by a car?

Each vehicle in the line is a state of the markov chain.

So, as stated in the problem, the probability of going from 'state truck' to 'state car' is 3/4 and the probability of going from 'state car' to 'state truck' is 1/5.

How do you get 4/9? I don't see that
Thank you!

Originally Posted by maria_stoeva

How do you get 4/9? I don't see that
Thank you!

One way is to examine the limit of T^n as n --> oo, where T is the transition matrix (in practice, just raise T to a large number like n = 30 and deduce the limit).

Alternatively, an approach using eigenvalues can be used.