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

Mar 17, 2015

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.

$\left(\begin{matrix}\null & A & B & C & D & F \\ M r s . X & 5 & 9 & 14 & 4 & 6 \\ M s . Y & 9 & 4 & 12 & 6 & 2 \\ M r . Z & 4 & 16 & 8 & 4 & 4\end{matrix}\right)$

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:
$P \left(A | B\right)$ which is read as "the probability of A given B".

For our example we are asked for
$P \left(F | M r s . X\right)$

Although there are 107 students recorded in our table, only 38 of them are in Mrs. X's class and
$P \left(F | M r s . X\right) = \frac{6}{38}$

Of course, the problem may be more complex and we might be asked for
$P \left(A \mathmr{and} B | M r s . X \mathmr{and} M r . Z\right)$
but the methodology of solution remains simple.