Question

If the wheel is broken, there is a chance it's not broken to give us blacks all the time. Maybe it's programmed to give white now, right when we are asked? In this case the correct answer to this - Impossible to evaluate probability since we are unaware of the nature of the wheel's faulty behaviour (if we suppose there is one)?

As a philosopher seeking truth through reasoned inquiry, how should I evaluate this scenario? Should I trust the mathematically calculated 1/2 odds, or does the extremely improbable empirical result suggest the wheel is biased? Is it possible to integrate these perspectives?

More broadly, how do we address the tension between conceptual probability models and observations of real-world randomness that seem to defy the odds?

Answer 1

Theoretical probability is based on equally likely outcomes, while empirical probability is based on observed frequencies. The law of large numbers states that observed frequencies tend to converge to theoretical probabilities with a larger number of repetitions. Integrating both perspectives provides a more comprehensive understanding of probability and biases.

Step-by-step explanation:

In this scenario, it is important to distinguish between theoretical probability and empirical probability. Theoretical probability is based on the concept of equally likely outcomes, while empirical probability is based on observed frequencies in real-world experiments. The fact that a wheel may appear biased in a limited number of trials does not necessarily mean that it is programmed to give certain outcomes. It is possible that with a larger number of repetitions, the observed frequencies would converge to the theoretical probability. This is known as the law of large numbers.

When evaluating a scenario like this, it is important to consider both the theoretical probability and the observed empirical frequencies. The theoretical probability of an unbiased wheel would be 1/2, but if the empirical evidence consistently suggests a different frequency, it would be worth further investigation to determine if the wheel is indeed biased.

Ultimately, integrating both perspectives can provide a more comprehensive understanding of the tension between conceptual probability models and real-world observations. It allows for a more nuanced evaluation of the likelihood of certain outcomes and the potential biases that may exist.