Addition Rule of Probability

For two mutually exclusive events A and B, the probability that A or B occur in one trial is the sum of the individual probability of each event.

For example, consider rolling a standard six-sided die. Let A be the event that a 3 is rolled and B be the event that a 4 is rolled. The probability of A or B can be found by adding the individual probabilities. P(3 or4) = P(3)+P(4) = 16 + 16 ⇓ P(3 or4)= 26= 13 The formula above can be generalized to events that are not necessarily mutually exclusive. If events are overlapping, the probability of the common outcomes are counted twice in P(A)+P(B), so an adjustment is needed.

For example, consider rolling a standard six-sided die. Let A be the event that an even number is rolled and B be the event that a prime number is rolled.

Using the formula gives the probability that the result of the roll is even or prime. P(AorB)=P(A)+P(B)-P(AandB) ⇓ P(even or prime)= 12+ 12- 16= 56 This probability can be verified by accounting for the five outcomes that are even or prime: 2, 3, 4, 5, and 6.

Proof

Proving the Addition Rule of Probability

For mutually exclusive events, the Addition Rule of Probability is a postulate.

P(AorB)=P(A)+P(B)

Therefore, no proof will be given for mutually exclusive events. Now, consider non-mutually exclusive events A and B.

In the Venn diagram above, it can be seen part of event A does not overlap event B. That part is labeled a. Similarly, the part of event B that does not overlap event A is labeled b. Furthermore, the overlapping part – also known as the intersection — of both events is labeled c. Notation:& P(A⋂ B)= c Meaning:& The probability of eventsA & and B happening is c. Furthermore, in the diagram it can be also seen that a, b, and c are mutually exclusive. Therefore, the union of event A and event B should be considered.

Notation	Meaning
P(A)= a+c	The probability of A happening is a+c.
P(B)= b+c	The probability of B happening is b+c.
P(A⋃ B)=a+b+c	The probability of A happening or B happening is a+b+c.

Finally, the fact that P(A⋃ B)=a+b+c will be used to prove the Addition Rule of Probability for non-mutually exclusive events.

P(A ⋃ B)=a+b+c

IdPropAdd

Identity Property of Addition

P(A ⋃ B)=a+b+c+0

Rewrite 0 as c-c

P(A ⋃ B)=a+b+c+(c-c)

CommutativePropAdd

Commutative Property of Addition

P(A ⋃ B)=a+c+b+c-c

AssociativePropAdd

Associative Property of Addition

P(A ⋃ B)=( a+c)+( b+c)- c

SubstituteValues

Substitute values

P(A ⋃ B)= P(A)+ P(B)- P(A⋂ B)

The rule has been proven for non-mutually exclusive events.