Given events A and B, the probability of A
given B is written as P(A  B). This expands into a fraction:
In other words, it is the percentage of B that
is also in A.
Constructing a 2x2 crosstab
We first introduce the concept of complements.
Example
A = it is raining in Vancouver
Not A = it is not raining in Vancouver
If P(A) = 75%, then P(Not A) = 25%
Complement rule: P(Not A) = 1 – P(A)
It should be noted that some textbook write
P(not A) as .
Example
Suppose 42% of executives read The Wall Street
Journal, 8% read The Economist and 2.4% read both publications. Construct a 2x2
crosstab of this.

WSJ 
Not WSJ 
Total 
EC 
2.4 
5.6 
8 
Not EC 
39.6 
52.4 
92 
Total 
42 
58 
100 
Note that the 42% is in the total cell of the
Wall Street Journal and the 8% is in the total cell of The Economist. Since 42%
read the Wall Street Journal, this means that 58% do not. Similarly, since 8%
read The Economist, this means that 92% do not. These percentages are known as
marginals since they are found in the margins of the crosstab.
Note that the 2.4% is in the crosstab of the
Wall Street Journal and The Economist since 2.4% of executives read both
publications. The remaining percentages are calculated by basic arithmetic.
Here is what they represent:
·
The 5.6% is the
percentage of executives who read The Economist but not the Wall Street Journal
·
The 39.6% is the
percentage of executives who read The Wall Street Journal but not The Economist
·
The 52.4% is the
percentage of executives who read neither publication
We can use the crosstab to answer conditional
probability questions.
What is the probability an executive reads The
Economist given they read The Wall Street Journal?
What is the probability an executive reads The
Wall Street Journal given they read The Economist?
Wording issues
Method 1 – use the word “if” in place of the
word “given”
Executive example – reword P(WSJ  EC)
What is the probability an executive reads The
Wall Street Journal if they read The Economist?
Note that we simply replace the word “given”
with the word “if”.
Method 2 – subject/object approach
In the sentence: I like pasta, I is the
subject, like is the verb and pasta is the object. In the subject/object
approach to conditional probability, we want
P(object  subject).
Executive example – reword P(WSJ  EC)
What percentage of executives who read The
Economist also read The Wall Street Journal?
The subject in the above sentence is:
executives who read The Economist. (Technically, “who read The Economist” is an
adjective phrase modifying executives.) In this question, we are looking at
executives who read The Economist and want to determine the percentage of these
executives who also read The Wall Street Journal.
Note the difference between these two
questions:
What percentage of executives read The Wall
Street Journal but not the Economist?
In this question, we are looking at executives
as a whole. We want P(WSJ and not EC). From the crosstab, the percentage is
39.6%.
What percentage of executives who read the
Wall Street Journal do not read The Economist?
In this question, we are looking at executives
who read the Wall Street Journal and want to determine the percentage of these
executives who do not read The Economist.
It should be noted that P(EC  WSJ) = 5.71%
and P(Not EC  WSJ) = 94.29% sum to 100%. The reason is that if an executives
read The Wall Street Journal, either they read The Economist or not.