Probability Axioms: Explaining Their Definition

[Avichal]

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?

 

[CompuChip]

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.

 

[Avichal]

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

 

[economicsnerd]

Suppose you have a coin which will be tossed (so $S=\{\text{heads}, \text{tails}\}$), 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?"

 

[Avichal]

Q1) Does this seem like a plausible situation?

Yes.

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.

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.

 

[CompuChip]

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?

 

[D H]

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.

 

[Avichal]

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!