Which probability interpretation is most reasonable to use: relative frequency, subjective, or classical?
The probability your company's sales will exceed seven million dollars this year. Is this relative frequency?
The probability that the S&P will increase or decrease by at least 25 points in one day? Is this classical?
The probability that you will get a ticket if you drive 70 mph on the interstate between work and home this coming Tuesday. Is this relative frequency?
To determine which probability interpretation is most reasonable to use in a particular scenario, we first need to understand the definitions and principles behind each interpretation:
1. Relative Frequency Interpretation: This interpretation is based on the concept that the probability of an event happening is equal to the long-term relative frequency of that event occurring in repeated, identical trials. In other words, the probability is estimated by observing the number of times the event occurs in relation to the total number of trials conducted.
2. Subjective Interpretation: This interpretation reflects an individual's personal belief or degree of certainty about the likelihood of an event happening. It takes into account factors such as personal experiences, knowledge, and subjective judgment.
3. Classical Interpretation: Also known as "a priori" or "theoretical" probability, this interpretation is based on theoretical probability models or assumptions. It assumes that all outcomes in a sample space are equally likely, making use of formal mathematical calculations.
Now, let's analyze each of the given scenarios to determine which interpretation is most appropriate:
1. The probability your company's sales will exceed seven million dollars this year:
This scenario does not involve any repeated trials, making it unsuitable for the relative frequency interpretation. It also does not solely rely on personal belief or subjective judgment, as it can potentially be estimated based on historical data or industry trends. Therefore, a combination of historical data and subjective judgment may be used to assess the likelihood, indicating that a subjective interpretation might be most reasonable in this case.
2. The probability that the S&P will increase or decrease by at least 25 points in one day:
This scenario, similar to the previous one, does not directly involve repeated trials, making the relative frequency interpretation less applicable. Additionally, it may be challenging to determine equal likelihoods for all possible outcomes and calculate probabilities using formal mathematical methods. As a result, a more suitable interpretation in this case would be the subjective interpretation, as individuals well-versed in financial markets may form personal beliefs or forecasts about the likelihood of such market movements.
3. The probability that you will get a ticket if you drive 70 mph on the interstate between work and home this coming Tuesday:
In this scenario, we do have repeated identical trials (e.g., driving on the interstate) and can potentially collect data on the occurrence of ticket issuance under similar circumstances. Therefore, the relative frequency interpretation would be most appropriate to estimate the probability. By analyzing historical data on similar driving situations and calculating the ratio of receiving a ticket to total instances, we can make an informed estimation of the probability.
In summary, the most reasonable probability interpretation to use varies depending on the specific scenario and available information. The relative frequency interpretation is suitable when we can observe repeated trials, while the subjective interpretation considers personal beliefs and judgments. The classical interpretation assumes equal likelihoods for all outcomes under specific theoretical models.