2. Properties of probability

Addition theorem

The addition theorem is used to answer the following questions:

• What is the probability of one event or another event or both events occurring?

• What is the probability of event A or event B occurring?

The word "or" indicates addition of the individual probabilities. The answers to the above questions are different depending on whether the events are mutually exclusive or independent.

Mutually exclusive events do not have an intersection or common area. The individual probabilities are simply added to arrive at the answer. For mutually exclusive events:

• P(A or B) = P(A) or P(B) = P(A) + P(B)

• P(A or B or C or D) = P(A) + P(B) + P(C) + P(D)

For two independent events, the intersecting or common area must be subtracted or it will be included twice. (Refer to the Venn diagram in section 11.0).

Probability (A or B) = P(A) or P(B) = P(A) + P(B) – P(A * B)

For three independent events:

P(A or B or C) = P(A) + P(B) + P(C) – P(A * B) – P(A * C) – P(B *C) + P(A * B * C)

 

Multiplication theorem

The multiplication theorem is used to answer the following questions:

• What is the probability of two or more events occurring either simultaneously or in succession?

• For two events A and B: What is the probability of event A and event B occurring?

The individual probability values are simply multiplied to arrive at the answer. The word "and" is the key word that indicates multiplication of the individual probabilities. The multiplication theorem is applicable only if the events are independent. It is not valid when dealing with conditional events. The product of two or more probability values yields the intersection or common area of the probabilities. The intersection is illustrated by the Venn diagrams in section 11.0 of this chapter. Mutually exclusive events do not have an intersection or common area. The probability of two or more mutually exclusive events is always zero.

For mutually exclusive events:

• P(A) and P(B) = 0

For independent events:

• Probability (A and B) = P(A) and P(B) = P(A) * P(B)

For multiple independent events, the multiplication formula is extended. The probability that five events A, B, C, D and E occur is

    P(A) and P(B) and P(C) and P(D) and P(E) = P(A) x P(B) x P(C) x P(D) x P(E)

 

Total probability formula

Sometimes it is useful to split a random experiment in two or more disjoint events whose union is the sample space and use the conditional probabilities with respect to these cases.

For arbitrary events and in a probability space the following formula holds

To see this, note that with disjoint union, and therefore

.

The formula above can be generalized to more than two cases. Suppose that we have a partition of the sample space by events which means that for and

then

.

The latter formula is sometimes called formula of total probability.

Example 7. An urn contains 2 black and 5 white balls. Two balls are drawn without replacement. What is the probability to get a black ball in the second draw.

Solution. Let ``The second ball is black’’ and ``The first ball is black’’

We have and

Applying the formula we thus get