Marginal Probability

• Contingency table consists of rows and columns of two attributes at different levels with frequencies or numbers in each of the cells. It is a matrix of frequencies assigned to rows and columns.
• The term Marginal is used to indicate that the probabilities are calculated using a Contingency table(or Joint Probability table).

PROBLEM STATEMENT

• In a used car lot, 70% have air conditioning and 40% have a CD player. 20% of the cars have both. What is the probability that a car has a CD player, given that it has AC? i.e. P(CD | AC)

SOLUTION 1
Assuming 100 cars are parked in the used car lot, 70 have AC, 40 have CD and 20 have both AC and CD
P(AC) = 70, P(CD) = 40 and P(CD AND AC) = 20
Using rules of multiplicity with not independent events, P(CD AND AC) = P(AC) * P(CD | AC). Therefore P(CD | AC) = 20/70 = 2/7

SOLUTION 2
Plotting the Contingency table as shown below

Fill the numbers in each cell such that the sum across the row matches the row Marginal Total and the sum across the column matches the column Marginal Total, as shown below

From the snapshot above, we can see that 70 cars have AC out of which 20 have CD. Therefore the probability of car with AC and CD is 20/70 = 2/7


REFERENCES
https://www.greatlearning.in/great-lakes-pgpba/