How should one treat probability in taking a decision?

Question

Suppose, I have a machine that accurately gives me the probability of any event occuring.

It's obvious what to do when

1. The probabilities are 0.5 and 0.5 (Do nothing)
2. The probabilities are 0 and 1 (Remain an atheist)
3. The probabilities are 1 and 0 (Become a Muslim)

However, consider a more nuanced scenario. Say I obtained from that machine that the probability of Islam being true is 50.1%, and that the probability of Islam being false is 49.9%. What would you do in this situation? Would you become a Muslim or stay an atheist/agnostic?

Edit:

Some commenters have raised a very good point. Only taking actions based on probability is not prudent. You have to weigh the consequences too. "If there's a 40% chance of rain, it's still reasonable to take an umbrella, unless carrying the umbrella is an excessive burden." So, it seems like I am just heading towards a traditional Pascal's wager :-)

Answer 1

I object to the grounds of your question. It is impossible to calculate a probability for the truth of a religion. Even if you could allocate a probability, a probability alone is no basis for decision making. You need to take into account all of the applicable pros and cons associated with the choices you are presented with, weighing them against each other in terms of their importance to you and their likelihood.

Answer 2

I would first argue that there is no such thing, and that if a machine is trying to spit out a "true" probability of Islam being true, it would simply be wrong if it displayed 50.1, since the real answer should either be 0 or 1. Islam is or isn’t true. There's no in between.

Any “probability” of it being true or false is just a human guess that maps into getting into the thorny issue of when it’s considered justified or unjustified to believe in a statement like “Islam is true”.

There is no such thing as a “true” or “objective” probability, especially when it comes to existential matters such as the Islamic god existing.

Answer 3

It depends on subjective criteria. For example, you may decide that a difference of 0.2% is not significant to you, and conclude "do nothing". Or you might use a Pascal wager type argument. Or you can look at the utilities of each choice in some other way. The probability values alone are not enough to decide, you need to supply some decision function that will probably include some amount of subjectivity.

Answer 4

Say I obtained from that machine that the probability of Islam being true is 50.1%, and that the probability of Islam being false is 49.9%.

I would say that the machine has determined that, with all the available information, there is a 50-50 chance of Islam being true.

My first question is; What information was used by the machine to arrive at these probabilities? I want to determine for myself whether the methodology and use is appropriate.

If no flaw can be found, then this leaves you with two choices:

1. Find a fair coin, flip it, and make your choice or a random number generator that spits out numbers from 1 to 1000. Any number 499 or less and the choice is Islam is false. Otherwise, Islam is true.

2. Read the Koran, the Bible, and all other religions and make a choice.

The coin flip is MUCH easier.

Edit for umbrella and 40% chance of rain.

1. Get a random number generator that randomly spits out integers from 1 to 10.

2. Any number of 4 or less: Carry the umbrella. Otherwise go umbrella free.

The other choice is to use the probability information as basis for further analysis: A forty percent chance of rain may lead to checking current weather satellite information to determine whether this general probability for rain applies to my specific region.

Decisions based solely on probability are a last resort to be used when there is no ability to get further information necessary to give the decision more certainty.

A common method to deal with rain probabilities is to buy an inexpensive 6 pack of collapsible umbrellas and keep one in each vehicle, one at work, etc. That way there is no math involved in keeping dry.

Answer 5

Decision Analysis

Decision Analysis is a formal method for assigning probabilities to outcomes in the context of making decisions. Another search term would be decision-making with probabilities.

The idea is to compute the expected value or expected payoff for each course of action then take the decision based on the prospect of gain or loss. Then the expected values are mapped to a decision tree. I cannot find a good reference via simple keyword search.

Surfing, Stock Trading, Risk Adventures

In a surfing competition the surfer must intuitively decide whether to ride a breaking wave or pass up that wave for a better wave that might arrive soon. If the surfer does not catch the wave properly then she can "kick out" which means to abort the ride and wait for another wave. There is a time limit on the competition and the surfer wants to optimize the points scored by judges for taking their best rides in the limited time. This is a type of decision analysis that does not involve formal estimates of probability but rather the surfer is intuitively deciding expected payoffs and taking action in real time.

Some experts argue that no one can time or beat the stock market because all information is priced into the decisions unless insiders or others have some strategic advantage. However, stocks trade with price action and there do seem to be traders who are better than the others at making money from price action. I compare this to how surfers decide to ride or not ride a wave, and when to enter a trade, and when to kick out to cut a loss short, and when to hold a winning trade to let profits run on the price trend. The point is that human skill is a factor in decision making which influences the expected payoffs.

When negotiating contracts, the skill and know-how of an expert making a bid is part of the expected payoff or expected outcome. Hunters and fisherman would assess each other's skill at finding game or fish and go with the most skilled hunters or fisherman to earn their living if possible. Probability theory is a tool to perhaps help improve outcomes based on decisions. But it cannot replace intuition and skill in a rich model of human decisions.