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# Probability Theorems

The following theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.

Theorem 1: The sum of the probability of happening of an event and not happening of an event is equal to 1. P(A)+P(¯A)=1P(A)+P(A¯)=1

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to 0. P(ϕ)=0P(ϕ)=0

Theorem 3: The probability of a sure event is always equal to 1. P(A) = 1

Theorem 4: The probability of happening of any event always lies between 0 and 1. 0 < P(A) < 1

Theorem 5: If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.

P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B)

Also for two mutually exclusive events A and B, we have P( A U B) = P(A) + P(B)

### Bayes’ Theorem on Conditional Probability

Bayes’ theorem describes the probability of an event based on the condition of occurrence of other events. It is also called conditional probability. It helps in calculating the probability of happening of one event based on the condition of happening of another event.

For example, let us assume that there are three bags with each bag containing some blue, green, and yellow balls. What is the probability of picking a yellow ball from the third bag? Since there are blue and green colored balls also, we can arrive at the probability based on these conditions also. Such a probability is called conditional probability.

The formula for Bayes’ theorem is P(A|B)=P(B|A)⋅P(A)P(B)P(A|B)=P(B|A)·P(A)P(B)

where, P(A|B)P(A|B) denotes how often event A happens on a condition that B happens.

where, P(B|A)P(B|A) denotes how often event B happens on a condition that A happens.

P(A)P(A) the likelihood of occurrence of event A.

P(B)P(B) the likelihood of occurrence of event B.

### Law of Total Probability

If there are n number of events in an experiment, then the sum of the probabilities of those n events is always equal to 1.

P(A1)+P(A2)+P(A3)+….P(An)=1