Probability Laws
Two events A and B are called mutually exclusive if they have no outcomes in common; that is, A and B = impossible event (empty set).
Three or more events are called mutually exclusive if they are pairwise mutually exclusive; that is, no two of them have outcomes in common.
Axiomatic Definition. Probability P is a real-valued function defined on events of a sample space S, satisfying the following axioms:
Axiom 1. For any event A , P(A) >= 0 .
Axiom 2. P(S) = 1 .
Axiom 3. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) .
It follows from Axiom 3 that if A1, A2, ... An is a finite number of mutually exclusive events then
P(A1 or A2 or ... or An) = P(A1) + P(A2) + P(A3) + ... + P(An) .
Law of the complement: P(not A) = 1 - P(A) .
The Addition Law: P(A or B) = P(A) + P(B) - P(A and B) .
Example A. There are ten students in a group. Here are the statistics:
An x means the student is taking the course.
Name | Al | Bee | Cee | Dee | Eli | Felix | Gigi | Howard | Iliad | Jay |
Algebra | x | x | x | x | x | |||||
Biology | x | x | x | x |
Let A = "a student is taking Algebra", and B ="a student is taking Biology".
Given: P(A) = 0.5 , P(B) = 0.4 , and P(A and B) = 0.3 , find
a. P(not B) = 1 - P(B) = 1 - 0.4 = 0.6 .
b. P(A or B) = P(A) + P(B) - P(A and B) = 0.5 + 0.4 - 0.3 = 0.6 .
c. P[not (A or B)] = 1 - P(A or B) =
1 - 0.6 = 0.4 .
d. P(not A and not B) = 0.4 .
e. P(A and not B) = P(A) - P(A and B) = 0.5 - 0.3 = 0.2 .
f. P[not(A and B)] = 1 - P(A and B) = 1 - 0.3 = 0.7 .
g. P[not A or not B] = P(not A) + P(not B) - P(not A and not B) = 0.5 + 0.6 - 0.4 = 0.7 .
Example B. Given: P(A) =
0.3 , P(B) = 0.5 , and P(neither A nor B) = 0.4 . Find
a. P(A or B) = 1 - P(not A and not B) = 1 - 0.4 = 0.6 .
b. P(A and B) = P(A) + P(B) - P(A or B) = 0.3 + 0.5 - 0.6 = 0.2 .
c. P(A and not B) = P(A) - P(A and B) = 0.3 - 0.2 = 0.1 .
d. P(B and not A) = P(B) - P(A and B) = 0.5 - 0.2 = 0.3 .
e. P[not (A or B)] = 1 - P(A or B) = 0.4 .