# Help me, please

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• Sep 5th 2009, 01:49 PM
maria_stoeva
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!
• Sep 5th 2009, 04:00 PM
mr fantastic
Quote:

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.
• Sep 5th 2009, 04:03 PM
Sampras
Quote:

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?
• Sep 5th 2009, 04:16 PM
pedrosorio
Quote:

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.
• Sep 5th 2009, 05:53 PM
maria_stoeva
How do you get 4/9? I don't see that
Thank you!
• Sep 5th 2009, 05:58 PM
mr fantastic
Quote:

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.