Question

A taxi driver provides service in two zones of a city. Fares picked up in zone \(A\) will have destinations in zone \(A\) with probability \(0.6\) or in zone \(B\) with probability \(0.4\). Fares picked up in zone \(B\) will have destinations in zone \(A\) with probability \(0.3\) or in zone \(B\) with probability \(0.7 .\) The driver's expected profit for a trip entirely in zone \(A\) is 6 ; for a trip entirely in zone \(B\) is \(8 ;\) and for a trip that involves both zones is 12 . Find the taxi driver's average profit per trip.

Short answer

The taxi driver's average profit per trip is \(8.8\).

Step-by-step solution

Determine the probability of each type of trip

We are given the probabilities of each type of trip: - Zone A to Zone A: 0.6 - Zone A to Zone B: 0.4 - Zone B to Zone A: 0.3 - Zone B to Zone B: 0.7 We are not given the probability that the driver starts in A or B, so we need to determine those probabilities. Since the probability of each type of trip must add up to 1, we can use algebra to solve for the probabilities of starting in zone A or B: Let P(A) be the probability of starting in zone A and P(B) be the probability of starting in zone B. Then, P(A) + P(B) = 1. The combined probability of starting in A and ending in B is 0.4 and of starting in B and ending in A is 0.3. The total probability of an A-B crossing can be expressed as P(A-B) = P(A) * 0.4 + P(B) * 0.3. Since the only other type of trip is within a single zone, we can also express the total probability of an A-A crossing: P(A-A) = P(A) * 0.6 and the total probability of a B-B crossing: P(B-B) = P(B) * 0.7. We will use these expressions to solve for P(A) and P(B) in the next step.

Solve for P(A) and P(B)

To find P(A) and P(B), we will use the fact that the sum of all probabilities must equal 1, and substitute in our expressions from Step 1. This gives us: P(A-A) + P(A-B) + P(B-A) + P(B-B) = 1 [ P(A) * 0.6 ] + [ P(A) * 0.4 + P(B) * 0.3 ] + [ P(A) * 0.3 + P(B) * 0.3 ] + [ P(B) * 0.7 ] = 1 Simplifying the equation, we get: 0.6P(A) + 0.4P(A) + 0.3P(B) + 0.3P(A) + 0.3P(B) + 0.7P(B) = 1 1.3P(A) + 1.3P(B) = 1 Since P(A) + P(B) = 1, we can substitute P(A) = 1 - P(B) into the equation above to solve for P(B): 1.3(1 - P(B)) + 1.3P(B) = 1 1.3 - 1.3P(B) + 1.3P(B) = 1 1.3 = 1 This cannot be a valid result, so we must assume the taxi driver starts in Zone A and Zone B with equal probability, meaning P(A) = 0.5 and P(B) = 0.5.

Calculate the average profit per trip

Now that we have the probabilities of starting in Zone A and Zone B, we can calculate the expected profit for each type of trip as follows: - Expected profit for A-A trip: P(A-A) * profit = 0.5 * 0.6 * 6 = 1.8 - Expected profit for A-B trip: P(A-B) * profit = 0.5 * 0.4 * 12 = 2.4 - Expected profit for B-A trip: P(B-A) * profit = 0.5 * 0.3 * 12 = 1.8 - Expected profit for B-B trip: P(B-B) * profit = 0.5 * 0.7 * 8 = 2.8 The average profit per trip can be found by adding the expected profit of each type of trip: Average profit per trip = 1.8 + 2.4 + 1.8 + 2.8 = 8.8 So, the taxi driver's average profit per trip is 8.8.

Key concepts

Probability Models

Understanding probability models is crucial when calculating expected outcomes in real-world scenarios, such as determining a taxi driver’s profit per trip in different city zones. These models utilize probabilities to represent uncertain events, serving as the foundation for making informed decisions. For instance, in the exercise, the trips a taxi driver takes are classified into two zones, A and B. The probabilities of staying within a zone or crossing to another are provided (0.6 and 0.4 for zone A, 0.3 and 0.7 for zone B, respectively).

Using these probabilities, we set up a model to calculate the expected number of trips in each category, which is essential to computing the overall expected profit. Probability models not only account for the likelihood of each outcome but also require consideration of the associated gains or losses – in this case, the profit from each type of trip. The art of crafting these models lies in understanding the variables at play and using them to predict future outcomes, which is exactly why they are so indispensable in financial planning and business strategy.

Profit Optimization

Profit optimization is central to business operations and the heart of many decision-making processes. It involves the strategic analysis and adjustment of various factors to maximize profit margins. In a case similar to our exercise, the taxi driver strives to maximize profits by analyzing his earnings from trips within and between the zones.

To optimize profit, understanding the expected value from different actions or strategies is pivotal. Making decisions based on probability models, a taxi driver can determine the most lucrative routes or where to focus their time. The expected profit from each type of trip (6 for A-A, 12 for A-B or B-A, and 8 for B-B) needs to be weighted by the likelihood of each trip occurring. Then, by summing up these values, we arrive at an average or expected profit per trip, which aids the driver in deciding which areas might be more profitable to work in, effectively optimizing their earning potential.

Markov Processes

Markov processes are mathematical models used to predict the behavior of systems that undergo transitions from one state to another, with the key feature being that the next state depends only on the current state and not the sequence of events that preceded it. This concept, while not explicitly mentioned in the taxi driver example, is inherently tied to the setup of the problem. The likelihood of the driver being in zone A or B after a fare is a sort of Markov process, where the 'next' fare's starting zone depends only on the 'current' fare's destination zone.

In more complex systems, like urban transportation or customer purchasing behaviors, the use of Markov decision processes, a combination of Markov processes with decision making, allows for the optimization of sequential decisions in the face of uncertainty. Applied to our taxi scenario, if we further consider the driver’s choice regarding where to pick up the next fare or the timing, Markov decision processes can be used to optimize routing and scheduling to maximize long-term profits.