P(A) = 15/180 = 0.0833
P(not A ) = 165 / 180 = 0.9167
Complementary
events: For any given event, its
complement
contains all the points that are
not
in the
event.
There are two rules with complementary events:
•
Complementary events are always mutually exclusive events.
•
The probability of complementary events always adds to 1.
4.2
Conditional Probability
Conditional probability:
determining the probability of an event occurring, given another event has already
occurred.
EXAMPLE 5: Suppose we pick a tasting at random. What is the probability that our randomly selected
tasting has a “good” quality rating?
IE . Anything else
.
A =
event of interest
.
A
e
compliment pf A
EXAMPLE 6: Suppose we
knew
that the randomly selected tasting was a high priced beer. Then:
•
How many tastings are known to be from high priced beers?
•
Of these high priced beers, how many of them have a good quality rating?
•
What is the probability that a randomly selected tasting has a good quality rating, knowing that it is a
high priced beer?
B
:
33
Independent Events
:
Two events are independent if
the occurrence of one event has
no effect
on the probability of the occurrence
of the other event
EXAMPLE 7: We may be interested in knowing if high priced beers are more likely to receive a good quality
rating, or if these two events are
independent
.
We can
test for independence by examining the conditional probabilities:
First, determine the probability of a randomly selected tasting having a good quality rating:
Second, determine the conditional probability of a randomly selected tasting have a good quality rating,
given that it is known to be a high priced beer:
Finally, compare these two probabilities:
What happened?
B changed the probability of A
Mathematically, a Test for Independence can be expressed as:
Independent if p(A) stays the same even if B happens first
P
(
A

B
) =
P
(
A
)
A second Test for Independence (less intuitive, but still true) is: If two events (
A
and
B
) are
known
to be
independent, then
P
(
A
∩
B
) =
P
(
A
) ×
P
(
B
) is true.
The mathematical tests can be applied in reverse:
34
A = good quality rating , p ( A ) = 36/180 =0.20
A = good quality rating
.
P(A B ) = 15/60 =0.25
B = high price

P (A ) =/P(AB)
If P(A) = P(AB)
P(B) = P (BA)
P(A
B ) = P(A) * p(B)
Then A + B are independent + vice versa