Topic 19: Conditonal Probability and Tables

A two-way table, commonly called a matrix, organizes data which can be characterized two different ways. Data organized in a two-way table sometimes reveals overlooked relationships between the variables.

Consider the tables used in Example 18.6 from the previous Topic.

and

Now, we can coalesce both of these into one two-way table.

The table in this example could be used to calculate conditional probabilities. A conditional probability consists of finding the probability of an event A given that an event B has occurred. It is important to know whether it's the proportion of A's that occur among the B's, or vice-versa.

For example, using the table above, in crashes involving vehicles with 15 or more passengers, what is the probability of rollover?
Answer: 7 / 10 = 0.7

Similarly, for all rollover crashes, what is the probability they involve a vehicle which can carry 15 or more passengers?
Answer: 7 / 263 = 0.027

For specific events A and B, then "A or B" is the event that A occurs, or B occurs, or both A and B occur. This leads to the identity:
P( A or B ) = P(A) + P(B) - P(A and B).

This is more general that the formula in Topic 18, since P(A and B) will take care of events A and B not being disjoint.

Now, let's consider picking a card at random from a standard deck of cards. The probability that it is an ace is 4/52, since there are 4 aces in a deck. If we replace this card, shuffle, and choose another card at random, what is the probability of choosing an ace? Still 4/52. We say that these two events are independent.

In general, A and B are independent when P( A given B ) = P(A).