Probability Tables

Often used as a convenient way to describe the joint probabilities of events.

Ex: Probability of Shirt Returns

Recall:

Ex: Product returns

Suppose a website sells three types of shirts, each with different return rates:

$P(R|A_1)=.15$

• Shirt 1: 60% of overall shirt sales, 15% of purchases returned $P(A_1)=.6$
• Shirt 2: 30% of overall shirt sales, 25% of purchases returned $P(A_2)=.3$
• Shirt 3: 10% of overall shirt sales, 35% of purchases returned $P(A_3)=.1$

For a randomly selected shirt purchase, what is the probability
that the purchase is shirt 1 and is returned?

$P(A_1\cap R)=P(R|A_1)P(A_1)=(0.15)(0.6)=0.09=9\%$

Link to original

All that data can be cleanly shown in a table:

	Yes ($R$)	No ($R^{\complement }$)	Total
($A_1$)	0.090	0.510	0.600
($A_2$)	0.075	0.225	0.300
($A_3$)	0.035	0.065	0.100
Total	0.200	0.800	1

Ex: Counts of Shirt Returns

	Yes ($R$)	No ($R^{\complement }$)	Total
($A_1$)	180	1020	1200
($A_2$)	150	450	600
($A_3$)	70	130	200
Total	400	1600	2000

This table represents everything in the first example but with counts instead. We can reach the original table by dividing everything by 2,000.