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The following case study was reported in the article "Parking Tickets and Missing Women," which appeared in an early edition of the book Statistics: A Guide to the \(U n\) known. In a Swedish trial on a charge of overtime parking, a police officer testified that he had noted the position of the two air valves on the tires of a parked car: To the closest hour, one was at the one o'clock position and the other was at the six o'clock position. After the allowable time for parking in that zone had passed, the policeman returned, noted that the valves were in the same position, and ticketed the car. The owner of the car claimed that he had left the parking place in time and had returned later. The valves just happened by chance to be in the same positions. An "expert" witness computed the probability of this occurring as \((1 / 12)(1 / 12)=1 / 44\). a. What reasoning did the expert use to arrive at the probability of \(1 / 44\) ? b. Can you spot the error in the reasoning that leads to the stated probability of \(1 / 44\) ? What effect does this error have on the probability of occurrence? Do you think that \(1 / 44\) is larger or smaller than the correct probability of occurrence?

Step by step solution

01

Identify expert's reasoning

The expert concluded that the probability of the car owner leaving and returning to the same parking spot with the tire valves in the exact same position as \(1 / 44\) by assuming each hour on a clock (like a 12-hour clock) as a possible position for the valves. Hence, with two valves, the expert considered the probability of one valve being in a particular position as \(1 / 12\) and the probability of the second valve being in the same position as again \(1 / 12\). The combined probability is the product: \((1 / 12) * (1 / 12) = 1 / 144\).

02

Spot the error

The expert made a mistake by considering the position of each valve as independent of one another. Actually, once the position of the first valve is established, the position of the second one is no longer uniformly distributed. Moreover, the probability was miscalculated as \(1 / 44\), whereas the calculations give \(1 / 144\).

03

Correct Probability

A correct approach would have been to consider the two positions as one event. Meaning, the correct statement is 'What is the probability that in any random hour, both air valves point to exactly 1 o'clock and 6 o'clock position respectively?'. As there are 12 hours on a clock, the correct probability of such an event would be \(1 / 12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independence of Events

In probability theory, understanding the independence of events is crucial for accurate calculations. Independence refers to events that do not affect each other's outcomes. For example, if we roll two dice, the result of one die doesn’t influence the result of the other. In the context of the exercise, the expert incorrectly assumed that the positions of the two tire valves were independent events.
He treated each valve's position as if it were separately chosen without any relation to the other, which led him to erroneously multiply probabilities. In reality, once one valve is in a particular position, the possible positions of the second one are interdependent, especially when specific fixed positions like 1 o'clock and 6 o'clock are specified for both.
This misinterpretation led to an incorrect probability calculation, demonstrating why recognizing the interdependence or independence of events is essential.

Probability Calculation

Calculating probability involves determining how likely an event is to occur. This involves understanding the potential outcomes and how many ways an event can happen relative to the total number of possible outcomes. In our example, the expert performed the probability calculation by finding the chance of each valve independently landing in a specific position out of 12, resulting in a probability of \( \frac{1}{12} \) for each valve.
The mistake happened when these were wrongly combined into a joint probability of \( \frac{1}{144} \) instead of the supposed \( \frac{1}{44} \). Actually, each combination of positions for both valves must be treated as a distinct event. Furthermore, every possible set of positions (like 1 o'clock and 6 o'clock) must be considered a single probability event by itself, hence resulting in a single calculation of \( \frac{1}{12} \).
Correctly calculating requires finding the likelihood of their specific configuration rather than treating these positions as independent and compounding probabilities incorrectly.

Misinterpretation of Probabilities

Misinterpretation of probabilities often leads to flawed reasoning and incorrect conclusions. In probability theory, it's easy to misread the independence of events or the way probabilities combine. The expert’s assumption that the probability was \( \frac{1}{44} \) due to independent positioning was incorrect.
Such errors arise from overlooking key details like whether events are truly independent or treating distinct outcomes as separate from the total probable set. This caused a glaring error in resultant probability, misrepresenting the actual chance of coincidental positioning.
Understanding the context and specifics of probability events helps in avoiding misinterpretations. Ensuring clarity about joint probability versus individual event probability is crucial to deriving correct, meaningful numerical probabilities, fostering clearer judgment and decision-making.