Rules of Probability

There are six rules of probability using which the probability of any compound event involving arbitrary events A and B, can be computed.

Rule 1:

p(A ∪ B) =p(A) + p(B) – p(A ∩ B)
This rule is also called the inclusion-exclusion principle of probability.

This formula reduces to
p(A ∪ B) =p(A) + p(B)
if A and B are mutually exclusive, since p(A ∩ B) = 0 in such a case.

Rule 2:

p(A ∩ B) =p(A) * p(B/A) = p(B) * p(A/B)

where p(A/B) represents the conditional probability of A given B and p(B/A) represents the conditional probability of B given A.

(a) p(A) and p(B) are called the marginal probabilities of A and B respectively. This rule is also called as the multiplication rule of probability.

(b) p(A ∩ B) is called the joint probability of A and B.

(c) If A and B are independent independent events, this formula reduces to

p(A ∩ B)=p(A) * p(B).

since when A and B are independent

p(A/B) = p(A)

and p(B/A) = p(B)

i.e. the conditional probabilities become same as the marginal (unconditional) probabilities.

(d) If A and B are independent, then so are A and BC; AC and B and AC and BC.

(e) Condition for three events to independent:

Events A, B and C are independent iff

p(ABC)=p(A) p(B) p(C)
p(AB)=p(A) p(B)
p(AC)=p(A) p(C)  A, B, C are pairwise independent
p(BC)=p(B) p(C)

Note: If A, B, C are independent, then A will be independent of any event formed from B and C.

For instance, A is independent of B ∪ C.

Rule 3: Complementary Probability

p(A) = 1 – p(AC)

p(AC ) is called the complementary probability of A and p(AC) represents the probability that the event A will not happen.

p(A) = 1 – p(AC)

p(AC) is also written as p(A′)

Notice that

p(A) + p(A′) =1
i.e. A and A′ are mutually exclusion as well as collectively exhaustive.

Also notice that by De Morgan’s law since

AC ∩ BC = (A ∪ B)C
p(AC ∩ BC) =p(A ∪ B)C = 1 – p(A ∪ B)
i.e.  p(neither A nor B) = 1 – p(either A or B)

Rule 4: Conditional Probability Rule

Starting from the multiplication rule
p(A ∩ B) =p(B) * p(A/B)
by cross multiplying we get the conditional probability formula

p(A/B) = p(A ∩ B)/p(B)

By interchanging A and B in this formula we get

p(B/A) = p(A ∩ B)/p(A)

Rule 5: Rule of Total Probability

Consider an event E which occurs via two different events A and B. Further more, let A and B be mutually
exclusive and collectively exhaustive events. This situation may be represented by following tree diagram

Now, the probability of E is given by value of total probability as

P(E) =P(A ∩ E) + P(B ∩ E) = P(A) * P(E/A) + P(B) *(E/B)

This is called rule of total probability.
Sometimes however, we may wish to know that, given that the event E has already occurred, what is the probability that it occurred with A? In this case we can use Bayes Theorem given below.

Rule 6: Baye’s theorem

If E1, E2, E3 …… EN are ‘n’ mutually exclusive events in a sample space ‘S’ such that S = E1 ∪ E2 ∪ E3 …….EN.
If A is any arbitrary event in sample space then, probability of even Ei when A has already occurred i.e.,