Probability - I


   
 
Multiplication Rule of Probability
We have already proved that if two events A and B from a sample S of a random experiment are mutually exclusive, then
 
 
In this section, we examine whether such a rule exists, if '' is replaced by '' and '+' is replaced by 'x' in the above addition rule. If it does exist, what are the particular conditions restricted on the events A and B.
 
This leads us to understand the dependency and independency of the events.
 
Example:
 
A bag contains 5 white and 8 black balls, 2 balls are drawn at random. Find
 
a) The probability of getting both the balls white, when the first ball drawn, is replaced.
 
b) The probability of getting both the balls white, when the first ball is not replaced.
 
Suggested answer:
 
a) The probability of drawing a white ball in the first draw is . Since the ball is replaced, the probability of getting white ball in the second draw is also .
 
 
b) The probability of drawing a white ball in the first draw is . If the first ball drawn is white and if it is not replaced in the bag, then there are 4 white balls and 8 black balls. Therefore, the probability of drawing a white ball in the second draw = .
 
 
In this case, the probability of drawing a white ball in the 2nd draw depends on the occurrence and non-occurrence of the event in the first draw.
 
 
Independent Events
 
Events are said to be independent if the occurrence of one event does not affect the occurrence of others.
 
Observe in case(a) of above example,
 
The probability of getting a white ball in the second draw does not depend on the occurrence of the event on the first draw.
 
However in case(b), the probability of getting a white ball in the second draw depends on the occurrence and non - occurrence of the event in the first draw.
 
It can be verified by different example.
 
If A and B are two independent events, then
 
 
This is known as Multiplication Rule of Probability.
 
The converse is also true, that is if two events A and B associated with a random experiment are such that
 
 
then the two events are independent.
 
Two events are said to be dependent if the occurrence of one affects the occurrence of the other. In this case,
 
 
In the above example,
 
let
 
A = event that the outcome is a white ball in the first draw
 
B = event that the out come is a white ball in the second draw
 
In case (a),
 
 
\ The probability that both the balls drawn is white
 
 
 
In case (b),
 
 
(Since the ball after the first draw is not replaced)
 
P(B) = P (first draw is white and second draw is white)
 
+ P (first draw is black and second draw is white)
 
 
(both the balls are white)
 
 
Here P(A B) P (A). P(B) since the events are not independent.
 
Independent Experiment
 
Two random experiments are said to be independent if for every pair of events E and F, where E is associated with the first experiment and F is associated with the second experiment, the probability of simultaneous occurrence of E and F, when the two experiments are performed, is the product of the probabilities P(E) and P(F), calculated separately on the basis of the two experiments.
 
i.e,. P(EF)= P(E).P(F)
 
Example:
 
Probability of solving a specific problem independently by A and B are  respectively. If both try to solve the problem independently, find the probability that the problems be solved.
 
Suggested answer:
 
Let A be the event of A solving the problem.
 
Let B be the event of B solving the problem.
 
Then,
 
 
P (A not solving the problem)
 
 
P (B not solving the problem)
 
 
 
(Considering the experiments as independent, because A and B solve the problem independently)
 
P (both not solving the problem)
 
 
Probability that the problem can be solved
 
 
Note:
 
If A and B are independent, then
 
i) Ac and Bc are independent
 
ii) Ac and B are independent
 
iii) A and Bc are independent
 
  
     
   
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