To keep track of the location of its cabs, Zephyr Cab has divided a town into three zones: zone I, zone II, and zone III. Zephyr's management has determined from company records that of the passengers picked up in zone I, 40% are discharged in the same zone, 50% are discharged in zone II, and 10% are discharged in zone III. Of those picked up in zone II, 10% are discharged in zone I, 70% are discharged in zone II, and 20% are discharged in zone III. Of those picked up in zone III, 10% are discharged in zone I, 10% are discharged in zone II, and 80% are discharged in zone III. If the initial distribution vector for the location of the taxis is
What will be the distribution after all of them have made one pickup and discharge?

A) 25% in zone I, 44% in zone II, 31% in zone III
B) 31% in zone I, 44% in zone II, 25% in zone III
C) 31% in zone I, 25% in zone II, 44% in zone III
D) 44% in zone I, 25% in zone II, 31% in zone III

Question

To keep track of the location of its cabs, Zephyr Cab has divided a town into three zones: zone I, zone II, and zone III. Zephyr's management has determined from company records that of the passengers picked up in zone I, 40% are discharged in the same zone, 50% are discharged in zone II, and 10% are discharged in zone III. Of those picked up in zone II, 10% are discharged in zone I, 70% are discharged in zone II, and 20% are discharged in zone III. Of those picked up in zone III, 10% are discharged in zone I, 10% are discharged in zone II, and 80% are discharged in zone III. If the initial distribution vector for the location of the taxis is ​  ​
What will be the distribution after all of them have made one pickup and discharge?
​
A) 25% in zone I, 44% in zone II, 31% in zone III 
B) 31% in zone I, 44% in zone II, 25% in zone III 
C) 31% in zone I, 25% in zone II, 44% in zone III 
D) 44% in zone I, 25% in zone II, 31% in zone III

Quizplus · Accepted Answer

To solve this problem, we need to use matrix multiplication.

Let's first set up the initial distribution vector as a 3x1 matrix:

\begin{bmatrix} 0.15 \ 0.55 \ 0.3 \end{bmatrix}

This represents the fact that initially, 15% of the cabs are in zone I, 55% are in zone II, and 30% are in zone III (according to the given distribution vector).

Next, we need to set up a transition matrix that represents the probability of a cab moving from one zone to another. We can do this using the information given in the problem:

\begin{bmatrix} 0.4 & 0.5 & 0.1 \ 0.1 & 0.7 & 0.2 \ 0.1 & 0.1 & 0.8 \end{bmatrix}

Each row of the matrix represents the probability of a cab starting in a particular zone and ending up in each of the three zones. For example, the first row represents the probabilities of a cab starting in zone I and ending up in zone I, zone II, and zone III, respectively (40%, 50%, and 10%).

To calculate the distribution after one pickup and discharge, we need to multiply the initial distribution vector by the transition matrix:

\begin{bmatrix} 0.4 & 0.5 & 0.1 \ 0.1 & 0.7 & 0.2 \ 0.1 & 0.1 & 0.8 \end{bmatrix} 
\begin{bmatrix} 0.15 \ 0.55 \ 0.3 \end{bmatrix} 
= 
\begin{bmatrix} 0.31 \ 0.44 \ 0.25 \end{bmatrix}

This result tells us that after one pickup and discharge, 31% of the cabs will be in zone I, 44% will be in zone II, and 25% will be in zone III. Therefore, the answer is (B).

To Keep Track of the Location of Its Cabs, Zephyr