# Probability of equally likely events

• I
"Why 2 equally-likely events has each a probability of 0.5 ?"

If i explain this saying that is due to the frequences as N goes to infinity, i'm saying a tautology cause it's implied in the definition of probability.

So, going to a deeper level, why the probabilities of each of 2 equally-likely events is 0.5 ? It's simply because : "Nature behaves this way" or we can say more ?

andrewkirk
Homework Helper
Gold Member
When we say that N events are 'equally likely' we mean that they have the same probability. So if we know that exactly one of N specified events must occur, and the events are equally likely, we know (from the 'equally likely' bit) that they all have the same probability, call it P and (from the fact that one of them must occur, and that two or more cannot both occur) that the sum of the probabilities is 1, ie

P + P + .... (N times) ... + P = 1

So we have N x P = 1. That is, P = 1/N.

In the above case, N=2, so P=0.5.

If we then ask what 'probability' or 'likely' means, we are getting into deep waters that probability theorists and philosophers love to debate, principally because there are no answers.

Aleoa
Stephen Tashi
"Why 2 equally-likely events has each a probability of 0.5 ?"
If i explain this saying that is due to the frequences as N goes to infinity, i'm saying a tautology cause it's implied in the definition of probability.

The formal treatment of probability in mathematics (as a special case of "measure theory") says nothing about the actual frequency of an event. People who apply probability theory to problems will interpret probability theory in various ways and may attempt to say something about the observed frequency of events.

The usual interpretation of phrase "equally likely events" asserts they are assigned the same probability. If there are only two events in the set of outcomes, it is an axiom of probability theory that their probailities sum to 1. So, from p + p = 1, one may deduce p = 1/2.

The question you mean to ask is: "Why do 2 equally likely events that are the only two possible outcomes in a situation each have probability 1/2". The mathematical argument why this is so has nothing to do with the behavior of Nature.

People who apply probability theory to situations in Nature usually think of the probability of an event as guaranteeing (with certainty) that the frequency of the event in a large number of independent trials will be nearly equal to the probability of the event. This is not a result that can be proven from mathematical probability theory, although this type of thinking resembles the result called "The Law Of Large Numbers".

Aleoa
Dale
Mentor
2020 Award
So, going to a deeper level, why the probabilities of each of 2 equally-likely events is 0.5 ? It's simply because : "Nature behaves this way" or we can say more ?
I wouldn’t go to nature. I would reference the second Kolomgorov axiom. It follows from that.

Aleoa
Then we conjecture from experience that if we flip a coin for many many ( N ) times we get approximately N/2 heads. And from this conjecture we deduce the axiom and so the mathematical definition of probability. Is this correct ?

Stephen Tashi
Then we conjecture from experience that if we flip a coin for many many ( N ) times we get approximately N/2 heads. And from this conjecture we deduce the axiom and so the mathematical definition of probability. Is this correct ?

The history of how current mathematical probability theory was developed is long and complicated. It's true that it began with analyzing gambling, but it is not a simple as the story you told.

(Notice how you are assuming the coin is a "fair coin". What if the coin is slightly bent?)

The history of how current mathematical probability theory was developed is long and complicated. It's true that it began with analyzing gambling, but it is not a simple as the story you told.

(Notice how you are assuming the coin is a "fair coin". What if the coin is slightly bent?)

So I have to settle for the fact that it works like this?

Stephen Tashi
So I have to settle for the fact that it works like this?

It's not clear how you think things work.

In your original question, you ask about (two) "equally likely events". You have to realize the implications of using the terminology "equally likely". The consequences of using that terminology lead to a vocabulary exercise and the use of some mathematical axioms. That's how the answer to your question works in the current terminology and axiomatic structure of probability theory.

The question of why the mathematical axioms were invented is complicated. I think the general concept is that people desired to explain the actually observed frequencies of events - for example, how often a coin tossed 10 times will land heads. By one concept of a "fair" coin, it should always land heads 5 times out of 10. This is not what happens with actual coins. So to invent a mathematical theory that describes actual coins, the notion of "probability" was invented to embody the idea that a fair coin has a "tendency" to come up heads half the time, but has no absolute guarantee of actually doing that.

The mathematical probability theory that resulted is, in a sense, circular because probability theory only has theorems about probabilities. It provides no absolute guarantees about the actual frequency of events. For example, it can answer questions like "What is the probability that if a fair coin is tossed 10 times, 7 of the tosses will result in heads?". It can't answer questions like "If a fair coin is tossed 10 times, will 7 of the tosses result in heads?".

outside of quantum mechanics, probability is merely a statement of ignorance. If it was possible to know the exact mechanics of a specific coin flip, there would be no uncertainty in the outcome. With N events and no prior information that would indicate any event more likely than another then it is reasonable to assign 1/N as the probability of any single event.

mathman