The contingency (or crosstabs) table can be noted using symbols also:

where the symbols shown above mean the following:

• ∩ = intersection (or ‘AND’)
• P(Ā) = P(not A) = 1 − P(A)
• P(B̅) = P(not B) = 1 − P(B)
• P(A ∩ B) = P(A and B)
• P(Ā ∩ B) = P(not A and B)
• P(A ∩ B̅) = P(A and not B)
• P(Ā ∩ B̅) = P(not A and not B)

The Totals

In the above table, we can calculate the total (singular) probabilities:

• P(A) = P(A and B) + P(A and B̅)
• P(Ā) = P(Ā and B) + P(Ā and B̅)
• P(B) = P(A and B) + P(Ā and B)
• P(B̅) = P(A and B̅) + P(Ā and B̅)

These ‘singular’ probabilities are the overall odds of A or B happening (or not happening).
Note: The overall total (bottom right box) should always equal 1 (ie: 100%).

Inside Probabilities (Intersections)

The ‘inside’ of the table contains the ‘overlaps’ (intersections) between the categories:

• P(A and B) = the odds of both A and B occurring
• P(A and B̅) = the odds of A occurring and B not occurring
• P(Ā and B) = the odds of A not occurring and B occurring
• P(Ā and B̅) = the odds of A not occurring and B not occurring

The AND‘s can be calculated using the conditional probabilities (‘givens’):

• P(A and B) = P(A|B)×P(B) = P(B|A)×P(A)
• P(A and B̅)  = P(A|B̅)×P(B̅) = P(B̅|A)×P(A)
• P(Ā and B) = P(Ā|B)×P(B) = P(B|Ā)×P(Ā)
• P(Ā and B̅) = P(Ā|B̅)×P(B̅) = P(B̅|Ā)×P(Ā)

Example 17.1.1

Problem Setup: Let us examine the effectiveness of two social media marketing campaigns:

• Let’s call them campaign A and campaign B
• In each campaign, people are shown an ad
• If someone clicks on the link provided after looking at the ad, we say they ‘click through’

The percentage who click through on each ad is called the ‘click-through rate‘ (CTR):

• The CTR (click-through rate) for campaign A is 2%
• The CTR (click-through rate) for campaign B is 5%
• If someone has already viewed ad A, the CTR for campaign B rises to 15%

Question: What is the probability of someone clicking through after both ads?

You Try: Can you write out the probabilities above using ‘stats notation’?

Need Help? Click below to reveal the answers (if needed).

Calculating the AND:

We can now calculate the odds of someone clicking through after both ads:

P(A and B) = P(B|A) × P(A) = 0.15 × 0.02 = 0.003

It is also possible to calculate missing values in the table. We only need to know the following:

• one or two of the totals
• some of the inside probabilities

We can calculate the rest knowing that the ‘totals’ are the sums across the rows and down the columns.

Example 17.1.2

Problem Setup: Let us continue with the two marketing campaigns example…

Question: Can you set up the contingency (cross-tabs) table for this problem?

Solution: Let us first calculate the compliments for A and B. These will be the probabilities of people NOT clicking through after seeing the ad:

• Campaign A‘s probability that someone does NOT click through =  P(Ā) = 1−P(A) =1−0.02 = 0.98
• Campaign B‘s probability that someone does NOT click through =  P(B̅) = 1−P(B) =1−0.05 = 0.95

You try: Can you add the above values and the values in Example 17.1.1 to the table?

• We can use the fact that we sum across the rows and columns to determine the totals.
• We can work backwards and subtract values from the totals to get the missing values.
• Let’s try this by continuing with the ad campaign example.

Example 17.1.3

Problem Setup: Let us continue with the two marketing campaigns crosstabs (contigency) table…

Question: Can you calculate the missing values in the table we built in Example 17.1.2?

You try: Calculate the missing values where needed and complete the CLR crosstabs table.

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