By the definition of probability, the sum of the probabilities of all outcomes in a sample space is 1. For example, consider the sample space S of an experiment of rolling a die.
S={1,2,3,4,5,6}
Each possible outcome has a probability of 16. Because we have 6 possible outcomes, we need to sum 16 six times.
1/6+ 1/6+ 1/6+ 1/6+ 1/6+ 1/6= 1
We can see that the sum is 1. Now, we will find how this is related to the relative frequency in a contingency table. For example, consider a contingency table that shows information of students' involvement in sports.
Involved in Sports
Not Involved in Sports
Totals
Man
0.22
0.26
0.48
Woman
0.28
0.24
0.52
Totals
0.50
0.50
1
Note that each joint relative frequency in the contingency table represents the possible outcomes when asking a man or a woman if they are involved in sports. This is why their sum must be equal 1, which can be seen in the bottom-right cell.
0.22+0.26+0.28+0.24= 1
Moreover, if we add the marginal relative frequencies their sum must also be equal to 1. Again, it is because they represent the possible outcomes but focus on only one category.
Total Row:&0.50+0.50= 1
Total Column:&0.48+0.52= 1
Therefore, the sum of the probabilities and the sum of the relative frequencies is 1 since both represent the probability of each possible outcome. Please note that examples may vary.
