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. The statement and proof of “Addition theorem” and its usage in various cases is as follows.
Mutually exclusive events:
Two or more events are said to be mutually exclusive if they don’t have any element in common. i.e. if, the occurrence of one of the events prevents the occurrence of the others then those events are said to be mutually exclusive.
Example:
The event of getting 2 heads, A and the event of getting 2 tails, B when two coins are tossed are mutually exclusive.
Because A = {HH}; B = {TT}.
Mutually exhaustive events:
Two events are said to be mutually exhaustive if there is a certainty of occurring at least one of those two events. i.e. one of those events will definitely happen.
If A and B are two mutually exhaustive then the probability of their union is 1.
i.e. P(AUB)=1.
Example:
The event of getting a head and the event of getting a tail when a coin is tossed are mutually exhaustive.
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).
Example:
If the probability of solving a problem by two students George and James are 1/2 and 1/3 respectively then what is the probability of the problem to be solved.
Solution:
Let A and B be the probabilities of solving the problem by George and James respectively.
Then P(A)=1/2 and P(B)=1/3.
The problem will be solved if it is solved at least by one of them also.
So, we need to find P(AUB).
By addition theorem on probability, we have
P(AUB) = P(A) + P(B)- P(A∩B).
P(AUB) = 1/2 +.1/3 – 1/2 * 1/3 = 1/2 +1/3-1/6 = (3+2-1)/6 = 4/6 = 2/3
Note:
If A and B are any two mutually exclusive events then P(A∩B)=0.
Then P(AUB) = P(A)+P(B).
Do you also need help with Geography? Take a look at our Geography tutoring services.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Okotoks visit: Tutoring in Okotoks.
