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STAT 100A
Section 4.6: Independence, Conditional Probability, and the Multiplication Rule
Example: The probability of observing an even number
(event A) on a toss of a fair die is 0.5, where
S = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}.
Suppose we’re given the information that on a particular
throw of the die the result was a number less than or equal
to 3 (event B), where
B = {1, 2, 3}.
Would the probability of observing an even number on that
throw of the die still be equal to 0.5?
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Conditional Probability Formula
To find the conditional probability that event
A occurs given that event B occurs, divide
the probability that both A and B occur by
the probability that B occurs.
P(A | B) = __________
Revisit Dice Example:
P(A | B) = __________ = __________ =
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Multiplication Rule
P(A B) = P(B) . P(A | B)
or
P(A B) = P(A) . P(B | A)
Events A and B are independent events if
the occurrence of B does not alter the
probability that A has occurred.
P(A | B) = P(A) or P(B | A) = P(B)
Events that are not independent are said to
be dependent.
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Multiplication Rule for Independent Events
If events A and B are independent,
P(A B) = P(A) . P(B).
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Mutually exclusive events are never
independent events.
Two events cannot be both mutually
exclusive events and independent events.
