Question

A police car is on patrol in a neighborhood known for its gang activities. during a patrol, there is a 60% chance of responding in time to the location where help is needed; else regular patrol will continue. upon receiving a call, there is a 10% chance for cancellation (in which case normal patrol is resumed) and a 30% chance that the car is already responding to a previous call. when the police car arrives at the scene, there is a 10% chance that the instigators will have fled (in which case the car returns back to patrol) and a 40% chance that apprehension is made immediately. else, the officers will search the area. if apprehension occurs, there is a 60% chance of transporting the suspects to the police station; else they are released and the car returns to patrol. express the probabilistic activities of the police patrol in the form of transition matrix.

Answer 1

The probabilistic activities of the police patrol can be expressed with a transition matrix. The matrix represents the probabilities of transitioning between different states such as patrol, responding, arrived, transporting, and station. The transition matrix is constructed based on the given probabilities of each event occurring.

Step-by-step explanation:

To express the probabilistic activities of the police patrol in the form of a transition matrix, we need to consider the different probabilities of each event occurring. Based on the given information:

1. There is a 60% chance of responding in time to the location where help is needed. This corresponds to the transition from the current state (patrol) to the next state (responding).
2. If a call is received, there is a 10% chance of cancellation, in which case the car returns to regular patrol. This corresponds to the transition from the current state (responding) back to the previous state (patrol).
3. If a call is received, there is a 30% chance that the car is already responding to a previous call. This corresponds to the transition from the current state (patrol) to the next state (responding).
4. When the car arrives at the scene, there is a 10% chance that the instigators will have fled, in which case the car returns to regular patrol. This corresponds to the transition from the current state (arrived) back to the previous state (patrol).
5. When the car arrives at the scene, there is a 40% chance of apprehension, in which case the car transitions to the next state (transporting) with a probability of 60%, or returns to regular patrol with a probability of 40%.
6. If apprehension occurs, there is a 60% chance of transporting the suspects to the police station. This corresponds to the transition from the current state (transporting) to the next state (station).
7. If apprehension does not occur, the suspects are released and the car returns to regular patrol. This corresponds to the transition from the current state (transporting) back to the previous state (patrol).

With these probabilities in mind, we can construct the transition matrix for the police patrol activities:

Patrol

Responding

Arrived

Transporting

Station

Patrol

00.6000

Responding

0.300.100

Arrived

0000.40

Transporting

00000.6

Station

1010.60