Question

Suppose Adam guesses a number between 1 and 10 from a random number generator. Suppose Bethany guesses a number between 1 and 100 from a random generator.

The probability of Adam guessing the correct outcome is 1/10, whereas it’s 1/100 for Bethany.

Something about this seems to create the implication that it’s "harder" for nature to create the first event instead of the second. But is this misconstruing probabilities?

What is the probability of the event of Adam guessing the correct outcome occurring vs. the event of Bethany guessing the correct outcome? It is not as if there were 10 events with equal likelihood in a hat where one of them was Adam picking the correct number that nature decided to choose from. How does one then compare probabilities of events?

Answer 1

The probabilities of Adam guessing a number correctly (1/10) and Bethany guessing another correctly (1/100) represent their respective chances within their own outcome sets and are not directly comparative in terms of one being 'harder' for nature to create. These probabilities reflect the theoretical odds and will approach their expected frequencies with a large number of trials due to the law of large numbers.

Step-by-step explanation:

The concept of probability is often misunderstood, especially when comparing probabilities of different events. The likelihood of Adam guessing a number between 1 and 10 is 1/10, while Bethany guessing a number between 1 and 100 has a probability of 1/100. While it might seem that nature has a 'harder' time creating the first event compared to the second, this is not accurate.

Probabilities simply represent the odds or chances of a specific outcome within a defined set. Adam's and Bethany's scenarios involve different sets of outcomes: 10 possibilities for Adam and 100 for Bethany. It does not imply that nature 'chooses' one event over the other; it is just an expression of how likely an event is, relative to the number of possible outcomes.

Understanding that these probabilities are independent and theoretical helps to compare them without misconstruing their meaning. The law of large numbers states that as an experiment is repeated many times, the relative frequency of outcomes will approximate the theoretical probability. Hence, if Adam and Bethany were to make many guesses, we would expect Adam to guess correctly about 10% of the time while Bethany would guess correctly about 1% of the time.