SETTING UP A PROBABILITY TABLE



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):

R L

----------- -----------

S P(S ( R) P(S ( L) | P(S)

B P(B ( R) P(B ( L) | P(B)

----------- ----------- ------

P(R) P(L) | 1

Here is the probability table with the numbers:

R L

------ ------

S 0.56 0.03 | 0.59

B 0.14 0.27 | 0.41

------ ------ ------

0.70 0.30 | 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.

Men Women Boys Girls

-------------- -------------- -------------- --------------

Survived 332 318 29 27

Died 1,360 104 35 18

Here is the probability table that would be derived from that occurrence table:

Men Women Boys Girls |

-------------- -------------- -------------- -------------- | -------------

Survived 0.14935 0.14305 0.01305 0.01215 | 0.31759

|

Died 0.61179 0.04678 0.01574 0.00810 | 0.68241

-------------- -------------- -------------- -------------- | -------------

0.76113 0.18983 0.02879 0.02024 | 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 non-commutativeness 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.

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