Probability Axioms

1. Jul 31, 2013

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?

2. Jul 31, 2013

CompuChip

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. Jul 31, 2013

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

4. Jul 31, 2013

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?"

5. Aug 1, 2013

Avichal

Yes.
Yes, although I am a bit unsure what you are asking.
48%

Sorry but I couldn't find any relevance to my question earlier.

6. Aug 1, 2013

CompuChip

What would the probability of tails be for this rigged coin, using your definition above?

7. Aug 1, 2013

D H

Staff Emeritus
The relevance is that you used the third axiom to calculate that 48% figure.

8. Aug 1, 2013

Avichal

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