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

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How do you find the probability of A and B?
What is the probability that events A and B both occur?
What is the P A and B if events A and B are independent?
Where do I find P AUB independent events?

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).