Question

What is a conditional probability in a two-way table?

Answer 1

Conditional probability is the probability of an event given the probability of another event. With reference to a two way table it is most often the probability of an event identified by a column (or row) given an event identified by a row (or column).

This is most easily seen with an example:

Suppose students are taking Math classes from one of 3 teachers; students have recently taken a standardized test and received alphabetic grades. Counts of the number of students who received each possible grade within each class have been collected as displayed in the two-way table below.

`(( ,A,B,C,D,F),(Mrs. X, 5,9,14,4,6),(Ms. Y, 9,4,12,6,2),(Mr. Z, 4,16,8,4,4))`

One possible question that might be asked is
"What is the probability that a student randomly selected from Mrs. X's class got an F?"

Typical notation for conditional probability is of the form:
`P(A|B)` which is read as "the probability of A given B".

For our example we are asked for
`P(F|Mrs. X)`

Although there are 107 students recorded in our table, only 38 of them are in Mrs. X's class and
`P(F|Mrs.X) = 6/(38)`

Of course, the problem may be more complex and we might be asked for
`P(A or B | Mrs. X or Mr. Z)`
but the methodology of solution remains simple.