# How to rationally decide between two events A and B having happened

If someone listed you an event A and an event B, told you that only one of those events happened today, and then asked you which one would you bet on, how should one make this bet?

Initially, I was thinking that this bet should be based on the likelihoods of the event. But likelihoods depend on how you categorize the event.

If event A = The mayor was killed today and event B = John (who you don’t know) lost his shoes, how would you decide between them? Many people may interpret the probability of the latter event to be higher since the probability of a person losing his shoes is high in a given area and certainly has a higher frequency than mayors being killed. But the probability of John losing his shoes may be undefined or low and may be even lower than the mayor being killed that day. Perhaps John is the type to never lose his shoes.

So what would a rational decision of this bet look like? Or is there no way to know and is too dependent on many factors?

• What you're asking for is basically how to assign probabilities or, more broadly, if even probability-related magnitudes should be taken into account for rationally deciding over two events. This is particularly problematic for modeling fault detection with AI, where events are extremely unlikely yet really important to correctly detect without failiure. Most successful approaches use deep learning/neural networks for it, instead of probabilistic solutions.
– user64708
Commented Mar 3, 2023 at 10:02
• @eirene Non-trivial neural networks are effectively pseudo-randomly probabilistic solutions. They will make the same mistakes every time, and once you've discovered a class of inputs that gets miscategorised you can characterise and (usually) explain it, but it's very difficult to predict its failure modes. Commented Mar 4, 2023 at 4:16
• This type of problem is beyond the capacity of ordinary folk like myself. El rachum/Allah rahim. Commented Mar 4, 2023 at 10:10
• @wizzwizz4 In machine learning it is not the same a probabilistic model than a model whose output can be probabilistically modelled. A canonical example of probabilistic model is Naive-Bayes. With probabilistic solution I meant no probabilistic models are usually used as solution to the fault detection problem.
– user64708
Commented Mar 4, 2023 at 12:03

I'm not sure why the event being in the past matters here. You are asking in effect whether there is a rational way to make a betting choice based on partial information. Suppose we change the example to a future event. Is there a rational way to determine whether to bet on Australia or India to win their cricket match? There's all kinds of information you could take into account: previous match results, recent performance of the teams and players, the team selection, the weather, the state of the pitch, etc. There is no way to possess all the relevant data, and no single way to combine that data to give an answer. People just disagree.

The problem is not merely that we have insufficient data. Consider the problem of betting on whether the price of a stock or commodity will go up or down. A lot of money rides on this every day. There is a great deal of data on historical prices, volumes of trade, supply and demand, seasonal and geographical factors, monetary and fiscal policy, consumer behaviour, etc. Financial firms pay a small fortune to acquire this data and process it. But they still cannot reliably determine the direction of a price move. The closest they can reasonably expect is to be right slightly more often than wrong.

It is the same if we bet on past events. We have partial information and some kind of model of how things work that might motivate us to place a bet. But usually the data is incomplete and the model is never fully reliable, so there is no straightforward way to make the decision. This does not make it pointless to consider the data. There is a lot we can do with data, particularly in simple situations, and especially if we can perform controlled experiments. But if the data is rare and the model is hard to test, then we are left with an imponderable.

A typical person would decide based on a largely subconscious judgement of which event seemed the more likely. There will be cases were one event seems much more likely than another (eg John tossed a coin and it was a head, v the mayor went mad and shot ten people) and cases, such as the one you mentioned, which will be less clear. Whether it is rational to make a bet in every case depends on too many factors to say.

So what would a rational decision of this bet [event A vs. event B] look like? Or is there no way to know and is too dependent on many factors?

I think there is no way to know; a rational bet between the two is 50/50. The second event (the lost shoes) might be more common than the first (the assassination or accidental death of an elected official), but I do not know enough about either event to assign a probability to them.

From the example, we do not know whether the mayor has been ill, or whether John is an especially forgetful person. So there is no context, either from the example or from daily life.

Sorry, but I have to say "too many factors."