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Independence Property
Given two events, and , if ( ) ( ), then does not depend on , and so the
dependence formula reduces to:
( ) ( ) ( )
( ) ( ) ( )
This result is important, because it allows you to only have to remember the “and” rule for
dependent events. If the next event does not depend on the prior event, then the end probability is
just a product of the two individual probabilities.
Though the ideas presented above might at first seem confusing, you‟ll notice that the idea of
joint probabilities has not changed. The only new caution is to take care to acknowledge whether
the events are independent or not. We‟ll consider a few more examples below.
Example 5: The probability that a resistor and capacitor both fail in a portable electronic
device in the fifth year of use is 0.95%. The probability that the resistor fails is 1.22% and the
probability that the capacitor fails is 1%. Are the events independent If they are not
independent, what is the probability that the capacitor fails given that the resistor fails
SOLUTION:
Let
If the two events are independent, then the product of unconditional probabilities should give us
the provided joint probability.
We have that,
( )
( )
If they are independent events, then
( )
However, the joint probability under independence is 0.0122%, not 0.95%.
Thus,
( ) ( ) ( )
Statistics for Decision-Making in Business © Milos Podmanik Page 112