SETTING UP A PROBABILITY TABLE
A probability table is a row-and-column presentation of marginal and joint
probabilities. Refer to the Football problem (No. 1 of the Twelve Practice
Problems below), which deals with two types stance by player X (the rows), and
two types of plays (the columns).
Marginal probabilities are probabilities of single events: P(S), P(B), P(R) and
P(L). They are so-named because they appear in the right-hand and lower
margins of the table.
Joint probabilities are probabilities of intersections ("joint" means happening
together). They appear in the inner part of the table where rows and columns
intersect.
The lower right-hand corner always contains the number 1. Each row of the
table must add from left to right, and each column of the table must add from top
to bottom.
Here is the setup for the Football problem (No. 1 of the Twelve Practice
Problems):
S
B
R
----------P(S  R)
L
----------P(S  L)
P(B  R)
----------P(R)
P(B  L)
----------P(L)
|
P(S)
|
P(B)
-----1
|
Here is the probability table with the numbers:
S
B
R
-----0.56
L
-----0.03
0.14
-----0.70
0.27
-----0.30
|
0.59
|
0.41
-----1.00
|
Unions and conditional probabilities do not appear in the table itself, but they may
be computed from values in the table according to the following two formulas,
where A and B can be any two events:
P(A  B) = P(A) + P(B) - P(A  B)
(The addition rule
for unions)
"The probability of the union of two events is equal to the sum of their
probabilities less the probability of their intersection."
P(A | B) = P(A  B) / P(B)
"A conditional probability is equal to the probability of the intersection of
the two events divided by the probability of the condition." (B, the event
after the "bar," is the condition.)
Often an Occurrence Table is set up before a probability table is prepared. In
the Football problem, no occurrence information is provided, so there is no
occurrence table. Previous editions of the textbook, however, gave an
occurrence table on the Titanic disaster.
Survived
Men
-------------332
Died
1,360
Women
-------------318
104
Boys
-------------29
35
Girls
-------------27
18
Here is the probability table that would be derived from that occurrence table:
Survived
Died
Men
-------------0.14935
Women
-------------0.14305
Boys
-------------0.01305
Girls
-------------0.01215
0.61179
-------------0.76113
0.04678
-------------0.18983
0.01574
-------------0.02879
0.00810
-------------0.02024
|
|
|
|
|
|
|
------------0.31759
0.68241
------------1.00000
The probabilities are computed by dividing each value in the occurrence table by
2,223, the total number of observations. This is why there is always a "1" in the
lower right-hand corner of a probability table. This 2 x 4 table has eight
intersections (an eight-way partition), the sum of whose probabilities is 1.
Probability tables present probability information in a very efficient and effective
way, simplifying the computation of unions and, especially, conditional
probabilities. The conditional probability formula, P(A | B) = P(A  B) / P(B), can
be difficult to apply. First, mistakes can be made substituting the events in a
problem for the generic A and B. Second, it is easy to forget that the probability
that you divide by, P(B), must be the probability of the condition (the event after
the bar) in the conditional probability.
With a probability table, a conditional probability is always computed by dividing
an intersection (inner) probability by a marginal (outer) probability. Examples:
1.
P(Men | Survived) = 0.14935 / 0.31759
" | Survived" means that we are operating in the first row of the
table, whose row sum is 0.31759. In that row, the Men number is
0.14935. Therefore, 0.14935 / 0.31759 is the proportion of the
survivors who were men. Hence P(Men | Survived) = 0.47026.
2.
P(Survived | Men) = 0.14935 / 0.76113
" | Men" means that we are operating in the first column of the
table, whose column sum is 0.76113. In that column, the Survived
number is 0.14935. Therefore, 0.14935 / 0.76113 is the proportion
of men who survived. Hence P(Survived | Men) = 0.19622.
These two conditional probabilities are a good example of the noncommutativeness of the condition operation. While there are eight
intersections and eight unions, there are 16 conditional probabilities.
Try writing all 16 conditional probabilities in symbolic form and computing their
values. Use M, W, B and G for the types of people and S and D for their fates.