Question

state and prove addition therom of probability

Answer 1

The probability of happening an event can easily be found using the definition of probability. But just the definition cannot be used to find the probability of happening at least one of the given events. A theorem known as “Addition theorem” solves these types of problems.

Addition theorem on probability:

If A and B are any two events then the probability of happening of at least one of the events is defined as P(AUB) = P(A) + P(B)- P(A∩B).

Proof:

Since events are nothing but sets,

From set theory, we have

n(AUB) = n(A) + n(B)- n(A∩B).

Dividing the above equation by n(S), (where S is the sample space)

n(AUB)/ n(S) = n(A)/ n(S) + n(B)/ n(S)- n(A∩B)/ n(S)

Then by the definition of probability,

P(AUB) = P(A) + P(B)- P(A∩B).