## What are Addition and Multiplication Theorems on Probability?

### Addition and Multiplication Theorem of Probability

State and prove addition and multiplication theorem of probability with examples

**Equation Of Addition and Multiplication Theorem **

**Notations :**

- P(A + B) or P(A∪B) = Probability of happening of A or B

= Probability of happening of the events A or B or both

= Probability of occurrence of at least one event A or B - P(AB) or P(A∩B) = Probability of happening of events A and B together.

**(1) When events are not mutually exclusive: **If A and B are two events which are not mutually exclusive, then

P(A∪B) = P(A) + P(B) – P(A∩B)

or P(A + B) = P(A) + P(B) – P(AB)

For any three events A, B, C

P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C) – P(C∩A) + P(A∩B∩C)

or P(A + B + C) = P(A) + P(B) + P(C) – P(AB) – P(BC) – P(CA) + P(ABC)

**(2) When events are mutually exclusive: **If

*A*and

*B*are mutually exclusive events, then

n(A∩B) = 0 ⇒ P(A∩B) = 0

∴ P(A∪B) = P(A) + P(B).

For any three events A, B, C which are mutually exclusive,

P(A∩B) = P(B∩C) = P(C∩A) = P(A∩B∩C) = 0

∴ P(A∪B∪C) = P(A) + P(B) + P(C).

The probability of happening of any one of several mutually exclusive events is equal to the sum of their probabilities,

*i.e.*if A

_{1}, A

_{2}……… A

_{n}are mutually exclusive events, then

P(A

_{1}+ A

_{2}+ … + A

_{n}) = P(A

_{1}) + P(A

_{2}) + …… + P(A

_{n})

*i.e.*P(Σ A

_{i}) = Σ P(A

_{i}).

(3) **When events are independent : **If

*A*and

*B*are independent events, then P(A∩B) = P(A).P(B)

∴ P(A∪B) = P(A) + P(B) – P(A).P(B)

(4) **Some other theorems**

- Let A and B be two events associated with a random experiment, then

**Generalization of the addition theorem :**If A

_{1}, A_{2}……… A_{n}are n events associated with a random experiment, then

- Booley’s inequality : If A
_{1}, A_{2}……… A_{n}are n events associated with a random experiment, then

These results can be easily established by using the Principle of mathematical induction.

**Conditional probability**

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of A under the condition that B has already occurred and P(B) ≠ 0, is called the conditional probability and it is denoted by P(A/B).

Thus, P(A/B) = Probability of occurrence of A, given that B has already happened.

Similarly, P(B/A) = Probability of occurrence of B, given that A has already happened.

Sometimes, P(A/B) is also used to denote the probability of occurrence of A when B occurs. Similarly, P(B/A) is used to denote the probability of occurrence of B when A occurs.

### Multiplication Theorem Of Probability

- If
*A*and*B*are two events associated with a random experiment, then P(A∩B) = P(A).P(B/A), if*P*(*A*) ≠ 0 or P(A∩B) = P(B).P(A/B), if P(B) ≠ 0. **Extension of multiplication theorem:**If A

_{1}, A_{2}……… A_{n}are n events related to a random experiment, then

P(A_{1}∩A_{2}∩A_{3}∩ … ∩A_{n}) = P(A_{1}) P(A_{2}/A_{1}) P(A_{3}/A_{1}∩A_{2})……P(A_{n}/A_{1}∩A_{2}∩…∩A_{n−1}),

where P(A_{i}/A_{1}∩A_{2}∩…∩A_{i−1}), represents the conditional probability of the event , given that the events A_{1}, A_{2}……… A_{i}_{−}_{1}have already happened.**Multiplication theorems for independent events:**If

*A*and*B*are independent events associated with a random experiment, then P(A∩B) = P(A).P(B)*i.e.,*the probability of simultaneous occurrence of two independent events is equal to the product of their probabilities. By multiplication theorem, we have P(A∩B) = P(A).P(B/A). Since*A*and*B*are independent events, therefore P(B/A) = P(B). Hence, P(A∩B) = P(A).P(B).**Extension of multiplication theorem for independent events:**If A

_{1}, A_{2}……… A_{n}are independent events associated with a random experiment, then

P(A_{1}∩A_{2}∩A_{3}∩ … ∩A_{n}) = P(A_{1}) P(A_{2})..… P(A_{n}).

By multiplication theorem, we have

P(A_{1}∩A_{2}∩A_{3}∩ … ∩A_{n}) = P(A_{1}) P(A_{2}/A_{1}) P(A_{3}/A_{1}∩A_{2})……P(A_{n}/A_{1}∩A_{2}∩…∩A_{n−1})

Since A_{1}, A_{2}………A_{n-1}, A_{n}are independent events, therefore

P(A_{2}/A_{1}) = P(A_{2}), P(A_{3}/A_{1}∩A_{2}) = P(A_{3}),……, P(A_{n}/A_{1}∩A_{2}∩…∩A_{n−1}) = P(A_{n})

Hence, P(A_{1}∩A_{2}∩A_{3}∩ … ∩A_{n}) = P(A_{1}) P(A_{2})..… P(A_{n}).

**Probability of at least one of the n independent events: **If p

_{1}, p

_{2}……… p

_{n}be the probabilities of happening of

*n*independent events A

_{1}, A

_{2}……… A

_{n}respectively, then

### Total probability and Baye’s rule

**(1)** **The law of total probability: **Let

*S*be the sample space and let E

_{1}, E

_{2}……… E

_{n}be

*n*mutually exclusive and exhaustive events associated with a random experiment. If

*A*is any event which occurs with E

_{1}or E

_{2}or …or E

_{n}, then

P(A) = P(E

_{1}) P(A/E

_{1}) + P(E

_{2}) P(A/E

_{2}) + ….. P(E

_{n})P(A/E

_{n}).

**(2) Baye’s rule: **Let

*S*be a sample space and E

_{1}, E

_{2}……… E

_{n}be n mutually exclusive events such that

We can think of E

_{i}’s as the causes that lead to the outcome of an experiment. The probabilities P(E

_{i}), i = 1, 2, …..,

*n*are called prior probabilities. Suppose the experiment results in an outcome of event

*A*, where P(A) > 0. We have to find the probability that the observed event

*A*was due to cause E

_{i}, that is, we seek the conditional probability P(E

_{i}/A). These probabilities are called posterior probabilities, given by Baye’s rule as