Understanding Probability Laws
Let a random experiment have sample space S.
Any assignment of probabilities to events must satisfy three basic laws of
probability, called
Kolmogorov’s Axioms
:
1)
For any event A, P(A) ≥ 0.
2)
P(S) = 1.
3)
If A and B are two mutually exclusive events (i.e., they cannot happen simultaneously), then P(A
∪
B) = P(A) +
P(B).
There are other laws in addition to these three, but Kolmogorov’s Axioms are the foundation for probability theory.
To achieve an understanding of the laws of probability, it helps to have a concrete image in mind.
I hope the following
examples will help.
Consider a single roll of two dice, a red one and a green one.
The table below shows the set of outcomes in the sample
space, S.
Each outcome is a pair of numbers--the number appearing on the red die and the number appearing on the green
die.
The event that consists of the whole sample space is the event that some one of the outcomes occurs.
This event is
certain to happen; if we roll the dice, the outcome cannot be something other than one of the 36 outcomes listed in the table.
Therefore, the probability associated with the event S is
P(S) = 1.
Number on
Green Die
1
(1, 1)
(2, 1)
(3, 1)
(4, 1)
(5, 1)
(6, 1)
2
(1, 2)
(2, 2)
(3, 2)
(4, 2)
(5, 2)
(6, 2)
3
(1, 3)
(2, 3)
(3, 3)
(4, 3)
(5, 3)
(6, 3)
4
(1, 4)
(2, 4)
(3, 4)
(4, 4)
(5, 4)
(6, 4)
5
(1, 5)
(2, 5)
(3, 5)
(4, 5)
(5, 5)
(6, 5)
6
(1, 6)
(2, 6)
(3, 6)
(4, 6)
(5, 6)
(6, 6)
1
2
3
4
5
6
Number on Red Die
If the dice are fair, then each of the 36 possible outcomes is equally likely.
Also the 36 possible outcomes are mutually