# Please explain the law of total probability?

Gaurav Sharma
19 Points
15 years ago

LAW OF TOTAL PROBABILITY:

The law of total probability is implied to the events that are mutually not exclusive. We can define this law as:
If n events A1, A2,……, An are n mutually exclusive events, then the probability of the happening of at least one of the events is the sum of the probabilities of the individual events. In symbols,

P [ A1+A2+A3+……+An ] = P[A1] + P[A2] + P[A3] + …… + P[An]

Or,

P [ A1 U A2 U …… U An ] = P[A1] + P[A2] + P[A3] + …… + P[An]

Now, we discuss the law of total probability in details:

Let N be the total number of mutually exclusive, exhaustive and equally likely events out of which m1 cases are favourable to A1, m2 cases are favourable to A2 and so on.

The probability of occurrence of the event A1 = P[A1] = m1/N

The probability of occurrence of the event A2 = P[A2] = m2/N
Similarly, for A3, A4, A5, ………

The probability of occurrence of the event An = P[An] = mn/N

The events being mutually exclusive and equally likely, the total number of cases favourable to the event A1 or A2 or … or An is
m = m1 + m2 + m3 + …… + mn
Thus,
P [ A1+A2+A3+……+An] = m/N

= ( m1 + m2 + m3 +…… + mn ) / N

= (m1/N) + (m2/N) + …… + (mn/N)

= P[A1] + P[A2] + P[A3] + …… + P[An]

Hence, P [ A1 + A2 + …… + An ] = P[A1] + P[A2] + …… + P[An].

This is the law of total probability.

Alternatively, the law can be understood or interpreted as follows:

If an event A consists of n mutually exclusive forms A1, A2, ……, An so that A happens whenever any one of the these events happens and conversely, then

A = A1+A2+A3+……+An and P[A] = P[A1] + P[A2] + P[A3] + …… + P[An].

The Law of Total Probability allows us to evaluate probabilities of events that are difficult to obtain alone, but become easy to calculate once we condition on the occurrence of a related event. Initially, we assume that the related event occurs, and subsequently, that it does not occur. The resulting conditional probabilities help us compute the total probability of occurrence of the event of interest.
The limitation of the law of total probability is that it cannot be used to evaluate the probabilities of the events that are not mutually exclusive.