by Addition Theorem and Multiplication Theorem are important rules in probability that help determine the probability of combined events. The Addition Theorem is used when finding the probability that at least one of two events occurs, while the Multiplication Theorem is used when finding the probability that two events occur together. These theorems are widely used in statistics, business forecasting, insurance, risk analysis, and decision-making.
Addition Theorem of Probability
Addition Theorem is used to calculate the probability of the occurrence of either one event or another event or both events.
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)
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).
Multiplication Theorem on Probability
Multiplication Theorem is used to calculate the probability that two events occur together.
If A and B are any two events of a sample space such that P(A) ≠0 and P(B)≠0, then
P(A∩B) = P(A) * P(B|A) = P(B) *P(A|B).
Example: If P(A) = 1/5 P(B|A) = 1/3 then what is P(A∩B)?
Solution: P(A∩B) = P(A) * P(B|A) = 1/5 * 1/3 = 1/15
Multiplication Theorem for Independent Events
When the occurrence of one event does not affect the occurrence of another event, the events are called independent events.
Two events A and B are said to be independent if there is no change in the happening of an event with the happening of the other event.
i.e. Two events A and B are said to be independent if
P(A|B) = P(A) where P(B)≠0.
P(B|A) = P(B) where P(A)≠0.
i.e. Two events A and B are said to be independent if
P(A∩B) = P(A) * P(B).
Example:
While laying the pack of cards, let A be the event of drawing a diamond and B be the event of drawing an ace.
Then P(A) = 13/52 = 1/4 and P(B) = 4/52=1/13
Now, A∩B = drawing a king card from hearts.
Then P(A∩B) = 1/52
Now, P(A/B) = P(A∩B)/P(B) = (1/52)/(1/13) = 1/4 = P(A).
So, A and B are independent.
[Here, P(A∩B) = = = P(A) * P(B)]
Note:
(1) If 3 events A,B and C are independent the
P(A∩B∩C) = P(A)P(B)P(C).
(2) If A and B are any two events, then P(AUB) = 1-P(A’)P(B’).
