In the test for independence, the claim is that the row and column variables are independent of each other. This is the null hypothesis.
The multiplication rule said that if two events were independent, then the probability of both occurring was the product of the probabilities of each occurring. This is key to working the test for independence. If you end up rejecting the null hypothesis, then the assumption must have been wrong and the row and column variable are dependent.
All hypothesis testing is done under the assumption

The test statistic used is the same as the chisquare goodnessoffit test. The principle behind the test for independence is the same as the principle behind the goodnessoffit test. The test for independence is always a right tail test.
In fact, you can think of the test for independence as a goodnessoffit test where the data is arranged into table form. This table is called a contingency table.
The test statistic has a chisquare distribution when the following assumptions are met
The following are properties of the test for independence
There is a program called CONTING (for contingency table) for the TI82 which will compute
the test statistic for you. You still need to look up the critical value in the table.