How do you define "Probability" without using a synonym?

[Myriad]

The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run.

Respectfully,
Myriad

 

[Perpetual Student]

I was just hoping that probability could be generally defined in purely mathematical terms just as it can be in the Bernoulli Trial case. I guess the best that can be done is to use some generalized formula.

If that's all you want, just call it:

The ratio of the occurrence of a specific outcome to all possible outcomes.

 

[jt512]

I would have thought that "probability" was like the word "charge" in which it could be the thing or the measure/quantity of the thing.

I suppose that the word "probability" can be used in a colloquial sense to mean a qualitative likelihood or degree of certainty, but you were asking for a definition of "probability" in its mathematical sense. Our degree of confidence in a belief is not obviously quantitative. A rigorously justified quantification of subjective degree of confidence was not worked out until the mid-20th century (to the best of my knowledge). So "probability" in its mathematical sense is a quantification of a concept that isn't obviously quantitative, and furthermore, it is a specific quantification—a scale ranging from 0 to to 1.

I was just hoping that probability could be generally defined in purely mathematical terms just as it can be in the Bernoulli Trial case.

It can and it is. But the problem is that probability is an abstract concept, and according, abstract mathematics is needed to describe it.

 

[Dave Rogers]

If that's all you want, just call it:

The ratio of the occurrence of a specific outcome to all possible outcomes.

The problem with that is the definition of the word "occurrence" as a variable that can take a continuous range of values, which is very different to the commonly used definition. In effect, you're using "occurrence" as, in part, a synonym for "probability".

Dave

 

[Perpetual Student]

The problem with that is the definition of the word "occurrence" as a variable that can take a continuous range of values, which is very different to the commonly used definition. In effect, you're using "occurrence" as, in part, a synonym for "probability".

Dave

Nonsense! The words occurrence and probability and quite apart from synonymous. There is no sense in which the word occurrence takes on a continuous range of values -- there is nothing preventing an interpretation of discrete values in my sentence.

Addendum: For example, consider a die:
The probability of rolling 3 is the ratio of the occurrence of 3 to the occurrence of all the numbers.

 

[Dave Rogers]

Nonsense! The words occurrence and probability and quite apart from synonymous. There is no sense in which the word occurrence takes on a continuous range of values -- there is nothing preventing an interpretation of discrete values in my sentence.

Addendum: For example, consider a die:
The probability of rolling 3 is the ratio of the occurrence of 3 to the occurrence of all the numbers.

Discrete or continuous is not the point. The point is that the common definition of the word "occurrence" is "that which occurs", which does not allow any range of values; something either is, or is not, an occurrence, according to the common definition. Using that definition, your addendum would of course be nonsensical; so the definition of the word "occurrence" you've used is clearly more complicated than that. In fact, the definition of "occurrence" you've used in your addendum is "the number of times a certain outcome occurs in a given number of repetitions of the process", which - going back to the OP - is undefinable for a process that occurs only once. So, again going back to the OP, you can only define "occurrence" for a one-off event in terms of the probability of a certain outcome occurring, which means that the definition is circular. Hence, your definition of "occurrence", while not synonymous with "probability", is dependent upon its definition.

Dave

 

[psionl0]

The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run.

Respectfully,
Myriad

I like this definition. "Expectation" is definitely not a synonym for probability.

I think I will use this definition with my students.

 

[psionl0]

If that's all you want, just call it:

The ratio of the occurrence of a specific outcome to all possible outcomes.

Works for discrete cases but for definition purposes, you would have to assume that all the possible outcomes were "equally likely".

 

[jt512]

I like this definition. "Expectation" is definitely not a synonym for probability.

I think I will use this definition with my students.

Defining probability in terms of expectation is circular, since expectation is defined in terms of probability.

What grade level are you students? What's their math background?

 

[jt512]

Works for discrete cases but for definition purposes, you would have to assume that all the possible outcomes were "equally likely".

You're right that his definition only works for the
discrete case, but it does not depend on all possible outcomes being equally likely.

 

[psionl0]

Defining probability in terms of expectation is circular, since expectation is defined in terms of probability.

Obviously! However, it technically meets the requirements of the OP and defining both in terms of money adds an interesting dimension for students.

What grade level are you students? What's their math background?

ATM I choose to deal only with senior high school students studying level 3 mathematics (the highest level). Not only is competition for such students low but ego is less of a problem for these students.

I believe that I could also be successful with primary school children but I am less interested in making a game out of everything and there is much more competition for primary school students (teachers making money on the side etc.)

 

[psionl0]

You're right that his definition only works for the
discrete case, but it does not depend on all possible outcomes being equally likely.

If you are defining probability merely in terms of counting outcomes, then all outcomes must be equally likely. Otherwise, the definition of probability becomes much more unwieldy.

 

[Startz]

The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run.

Respectfully,
Myriad

I think this is one of the accepted definitions. A slight modification, which avoids the issue of "long run," is

The probability of an event is the fraction of a dollar you should stake to win a dollar payoff to make a fair bet.

("Fair bet" may be easier than "long run.")

 

[Meed]

This thread may speak more to the difficulty of defining words in general than anything else.

 

[jt512]

[J]ust call it [t]he ratio of the occurrence of a specific outcome to all possible outcomes.

[Y]ou would have to assume that all the possible outcomes were "equally likely".

[I]t does not depend on all possible outcomes being equally likely.

If you are defining probability merely in terms of counting outcomes, then all outcomes must be equally likely. Otherwise, the definition of probability becomes much more unwieldy.

I believe you misinterpreted Perpetual Student's suggestion. He did not mean "counting outcomes," at least not in the sense you are using that ambiguous phrase. For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3.

Jay

 

[psionl0]

For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3.

I think you just highlighted my point. (But I get it).

 

[psionl0]

("Fair bet" may be easier than "long run.")

I have found that when my kids say, "it's not fair!" I can usually shut them up by asking them what they mean by "fair".

 

[Perpetual Student]

I believe you misinterpreted Perpetual Student's suggestion. He did not mean "counting outcomes," at least not in the sense you are using that ambiguous phrase. For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3.

Jay

Exactly.
If we consider the universe of possible outcomes of some event, it makes no difference if possible outcomes are equally likely or not. Simply take the ratio of the number of occurrences of interest to the universe of all outcomes. If the die were loaded, repeated experiments would give us our answer in any case for that particular die.

 

[Finn McR]

My math ran out before formal set theory, but isn't that the way to go? The probability of an outcome "A" is the ratio of the ways outcome "A" can occur to the universe of all possible outcomes. In basic probability problems there are many simplifying assumptions. Take die rolling. Most statistics problems assume that the probability of any given number 1-6 coming up in the roll of a d6 is 1/6. That ignores what one would consider the die "coming up" or "showing." What if it is cocked against another die, the side of a box that the die is rolled in, character sheets, pencils, D&D manuals on the table, etc. Not to mention my friend's cat that would occasionally pounce on a die and and whack it into some dark corner. Bemoaning dismissiveness as an easy-out is a bit dismissive in itself. The probability of an event A that satisfies conditions of outcome "A" IS the ratio of the outcomes satisfying "A" to all possible outcomes.