Question

Note the word "existence" in the question where I’m trying to be careful with my wording here. This can be better illustrated with an example.

Take the example of the cheating process. Suppose one observes that John has won four straight lotteries, each of which only has a 1 in a 10 million chance of winning. This obviously seems to make it likely that John cheated. If H = John cheated and F = John won by chance, it seems obvious that H is more likely.

But there is additional information here that makes it more obvious that John cheated apart from the fact that his winnings were improbable. For starters, we already know that people cheat and have an incentive to cheat. Secondly, we also know that cheating is possible as a mechanism. But what if we didn’t know this?

What if H instead was = cheating in lotteries is possible. Does these series of observations make it more likely that H is true? Or must this H be independent of any observations or predictions?

If we didn’t know beforehand that cheating was possible and that people have cheated before, should we now believe that cheating as a process is more likely to be true after observing John win many lotteries?

Answer 1

Without prior knowledge of cheating methods or incentives, the observation of statistically improbable events, such as winning multiple lotteries, does not independently confirm the possibility of cheating. Past outcomes do not influence the probability of future events, and knowledge of cheating mechanisms should be considered separately.

Step-by-step explanation:

The question posed by the student involves understanding probability and inference in the context of lotteries and cheating. If we consider the hypothesis H to be "cheating in lotteries is possible," and we observe an event such as John winning four straight lotteries, each with a 1 in 10 million chance of winning, we might be tempted to believe that the occurrence of this event would make H more likely. However, in probability theory, unless we have prior knowledge that cheating can and does occur, we cannot infer from improbable events alone that cheating is definitely taking place.

Statistically improbable events do occur by chance, and without prior knowledge of the possibility of cheating, the observation of John winning multiple lotteries does not independently confirm that cheating is possible or has occurred. It is crucial to avoid the common misconception similar to the gambler's fallacy, where past outcomes are believed to influence the probability of future events, despite each event being statistically independent. Knowledge such as people having an incentive to cheat or the existence of mechanisms to cheat must be considered separately from the observation of the unlikely event when inferring the likelihood of dishonest actions or processes.