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 of both the given events. A theorem known as “Multiplication theorem” solves these types of problems. The statement and proof of “Multiplication theorem” and its usage in various cases is as follows.
Multiplication theorem on probability:
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
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’).
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