Probability Rules
Calculating Probabilities Using Contingency Tables
Learning Objectives
Use contingency tables to calculate probabilities.
Possible Calculations Using Contingency Tables:
Contingency tables give us the following values:
- ‘Singular’ probabilities: P(A), P(B), P(Ā), P(B̅), …
- ‘Joint’ probabilities: P(A and B), P(A and B̅), P(Ā and B), P(Ā and B̅), …
We can use these probabilities and our probability rule to calculate many probabilities. Ex:
- P(At least one event occurring):
- P(Neither event occurring):
- P(One event occurring given another event occurred):
Let us explore understanding multiple ways to use the table to calculate:
- P(A or B) = P(at least one of event A or B occurs)
- P(neither A nor B) = P(neither event A nor event B occurs)
Example 18.1.1
Problem Setup: Let us continue with the two marketing campaigns example (Example 18.1):
- P(A) = the click-through rate for ad campaign ‘A’
- P(B) = the click-through rate for ad campaign ‘B’
- P(Ā) = probability of people not clicking through after viewing ad ‘A’
- P(B̅) = probability of people not clicking through after viewing ad ‘B’
- The 2×2 (contingency) table for this example is below:
| A | not A | Totals | |
| B | 0.003 | 0.047 | 0.05 |
| not B | 0.017 | 0.933 | 0.95 |
| Totals | 0.02 | 0.98 | 1 |
Questions: Use the above table to answer the following:
- What percent of people click through on at least one of the ads?
- What percent of people click through on neither ‘A’ nor ‘B’?
Solution: Click below to reveal the answers (after trying it yourself):
To get the conditional/given probabilities:
- Take any of the AND‘s in our table and divide by an overall probabilities.
- Ex:
Example 18.1.2
Problem Setup: Let us continue using the table given in Example 18.1.1.
Questions: Calculate the conditional probabilities listed below:
- What percent of people who clicked through on ad ‘A’ then click through on ‘B’?
- Given someone doesn’t click through ‘A’, what is the probability they click through ‘B’?
- Out of those who do not click through on ‘B’, what percent click through on ‘A’?
- Who should the company running these ads target for ad ‘B’? Those who clicked through on ‘A’ or those who did not?
Instructions: Try the questions yourself. Once done, click below to reveal the answers for each part.
Think of what comes ‘first’ in this conditional probability. This event goes behind the line
- In this case, clicking through on ‘A’ comes first.
- So, we are looking for P(B|A)
- Ie:
- This was actually the probability we were given at the start of this example in the last section!
Not clicking through after viewing ad ‘A’ comes first for this conditional probability.
- Ie: We are looking for P(B| not A) = P(B| Ā)
Not clicking through after viewing ad ‘B’ comes first for this conditional probability. Ie:
- We are looking for P(A| not B) = P(A | B̅)
In questions 1 and 2, we examined the following:
- Percent who clicked on ‘B’ after they clicked on ‘A’= P(B|A) =15%
- Percent who clicked on ‘B’ after not clicking on ‘A’= P(B|Ā) =4.8%
- We can tell that those who clicked on ‘A’ first are quite a bit more likely to click on ‘B’
- So, the company should target those who already clicked on A.
- We will revisit this idea later in the course with ‘Chi-squared’ testing
Key Takeaways: Calculating Probabilities Using Contingency Tables
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