o_18ufhu7n51jij1pedr1jjeu19lpa.pdf

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4.7 Conditional Probability 119
PROBABILITY MATRIX
Geographic Location
Northeast
D
Southeast
E
Midwest
F
West
G
Finance A
.12 .05 .04 .07
.28
Industry
Type
Manufacturing B
.15 .03 .11 .06
.35
Communications C
.14 .09 .06 .08
.37
.41 .17 .21 .21
1.00
Solution
a.
P (B ƒ F) =
P (B ¨ F)
P (F)
= .11
.21 = .524
Determining conditional probabilities from a probability matrix by using the formula
is a relatively painless process. In this case, the joint probability, P (B ¨ F), appears
in a cell of the matrix (.11); the marginal probability, P (F), appears in a margin (.21).
Bringing these two probabilities together by formula produces the answer, .11 .21 = .524.
This answer means that 52.4% of the Midwest executives (the F values) are in manufacturing
(the B values).
b.
P (G ƒ C) =
P (G ¨ C)
P (C)
= .08
.37 = .216 >
This result means that 21.6% of the responding communications industry executives,
(C) are from the West (G).
c.
P (D ƒ F) =
P (D ¨ F)
P (F)
= .00
.21 = .00
Because D and F are mutually exclusive, P (D ¨ F) is zero and so is P (D F). The rationale
behind P (D ƒ F) = 0 is that, if F is given (the respondent is known to be located in
the Midwest), the respondent could not be located in D (the Northeast).
Independent Events
INDEPENDENT EVENTS X, Y
To test to determine if X and Y are independent events, the following must be true.
P(X ƒY) = P(X) and P(Y ƒ X) = P(Y)
In each equation, it does not matter that X or Y is given because X and Y are independent.
When X and Y are independent, the conditional probability is solved as a marginal
probability.
Sometimes, it is important to test a contingency table of raw data to determine whether
events are independent. If any combination of two events from the different sides of the
matrix fail the test, P(X Y) = P(X), the matrix does not contain independent events.
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DEMONSTRATION
PROBLEM 4.10
Test the matrix for the 200 executive responses to determine whether industry type
is independent of geographic location.