Probability Axioms: Explaining Their Definition

In summary, the axioms are meant to give intuition for the probability of an event, in a way that is more understandable than just dividing by total outcomes. The example used was of a coin that is rigged so that heads and tails are both equally likely. This helps us to understand the probability of an event more intuitively.
  • #1
Avichal
295
0
Axioms are: -
1) P(E) >= 0
2) P(S) = 1
3) P(E1 U E2 U ...) = P(E1) + P(E2) + ... if all are mutually exclusive

Why are the axioms defined in such a way? Why not this simple axiom: - Probability of an event is number of favorable outcomes divided by total number of outcomes?
 
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  • #2
Avichal said:
Probability of an event is number of favorable outcomes divided by total number of outcomes?

That would assume that all events are equally likely, which - in general - they are not.
If S = { today it rains, today it doesn't rain } then P(today it rains) is not 1 / |S| = 1/2. If that were true, replace " it rains" by "we all die in a meteor impact".
So what you do is assign a probability P(s) to every ##s \in S##. The axioms make sure that it matches our intuition.
 
  • #3
I am still not comfortable with the 3) axiom. It seems a bit indirect to me.
Suppose we toss a coin and we want to find the probabilities of heads and tails. Now P(H) + P(T) = 1 ... from 3)
Since both are equally probable both are equal and hence P(H) = P(T) = 1/2
It is all indirect. We could have directly said that out of two possibilities head or tail is one and thus it is 1/2
 
  • #4
Suppose you have a coin which will be tossed (so [itex]S=\{\text{heads}, \text{tails}\}[/itex]), and it's weighted so that the probability of heads is 52%.

Q1) Does this seem like a plausible situation?
Q2) Does it seem plausible that mathematics can inform ones decisions of which bets to take concerning this coin?
Q3) What do you think is a reasonable answer to: "What's the probability of tails?"
 
  • #5
economicsnerd said:
Q1) Does this seem like a plausible situation?
Yes.
economicsnerd said:
Q2) Does it seem plausible that mathematics can inform ones decisions of which bets to take concerning this coin?
Yes, although I am a bit unsure what you are asking.
economicsnerd said:
Q3) What do you think is a reasonable answer to: "What's the probability of tails?"
48%

Sorry but I couldn't find any relevance to my question earlier.
 
  • #6
Avichal said:
Why not this simple axiom: - Probability of an event is number of favorable outcomes divided by total number of outcomes?

What would the probability of tails be for this rigged coin, using your definition above?
 
  • #7
Avichal said:
economicsnerd said:
Q3) What do you think is a reasonable answer to: "What's the probability of tails?"
48%

Sorry but I couldn't find any relevance to my question earlier.
The relevance is that you used the third axiom to calculate that 48% figure.
 
  • #8
D H said:
The relevance is that you used the third axiom to calculate that 48% figure.

Thinking more about it I realized the importance of the 3rd axiom. Nice example to make me understand.
Many Thanks!
 

1. What are probability axioms?

Probability axioms are a set of fundamental principles that define the mathematical concept of probability. These axioms provide a foundation for understanding and calculating probabilities in various scenarios.

2. Why are probability axioms important?

Probability axioms are important because they allow us to make logical and consistent deductions about the likelihood of events occurring. They provide a framework for understanding randomness and uncertainty in the world around us.

3. What are the three axioms of probability?

The three axioms of probability are the additivity axiom, the non-negativity axiom, and the normalization axiom. The additivity axiom states that the probability of the union of two disjoint events is equal to the sum of their individual probabilities. The non-negativity axiom states that the probability of any event cannot be negative. The normalization axiom states that the probability of the entire sample space is equal to 1.

4. How do probability axioms differ from other mathematical axioms?

Probability axioms are unique in that they deal with uncertainty and randomness, whereas other mathematical axioms deal with certainties and absolutes. Probability axioms also have real-world applications in fields such as statistics, economics, and physics.

5. Can probability axioms be violated?

No, probability axioms cannot be violated. They are based on fundamental principles and are necessary for the mathematical concept of probability to make sense. If any of the axioms are violated, it would no longer be considered a valid probability measure.

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