# If a and b are events with p(a)=0.4, p(b)=0.2 and p(a and b)=0.1 . find the probability of a given b

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 .

### How do you find the probability of A and B?

Formula for the probability of A and B (independent events): p(A and B) = p(A) * p(B). If the probability of one event doesn't affect the other, you have an independent event. All you do is multiply the probability of one by the probability of another.

### What is the probability that events A and B both occur?

The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩ B). If Events A and B are mutually exclusive, P(A ∩ B) = 0.

### What is the P A and B if events A and B are independent?

If A and B are independent events, then the events A and B' are also independent. Proof: The events A and B are independent, so, P(A ∩ B) = P(A) P(B).

### Where do I find P AUB independent events?

Events A and B are independent if: knowing whether A occured does not change the probability of B. Mathematically, can say in two equivalent ways: P(B|A) = P(B) P(A and B) = P(B ∩ A) = P(B) × P(A).