any number from 2 to 12. The probability of each of these possibilities can be easily 

 displayed by means of the array whose elements are the sums of the row and column of the 

 array element as shown in Table 13. 



Table 13. Probabi 



lily distribution for the 



8uin of two numbers from 1 to 6. 





k 1 





1 



2 



3 



4 



5 



6 



1 



2 



3 



4 



5 



6 



7 



2 



3 



4 



5 



6 



7 



8 



3 



4 



5 



6 



7 



8 



9 



4 



5 



6 



7 



8 



9 



10 



5 



6 



7 



8 



9 



10 



11 



6 



7 



8 



9 



10 



11 



12 



The probability of occurrence of each element in the array is 1/36. It is seen that the 

 numbers 2 and 12 can be obtained by only one combination. Numbers 3 and 11 can be 

 obtained in two ways; 4 and 10 can be obtained in three ways; etc. The frequency of 

 occurrence of each number from 2 to 12 can be obtained by counting the number of 

 combioations yielding that number and dividing by 36. The results are shown in Table 14. 

 The cumulative frequency (i.e., the frequency that the number of spots will equal or exceed 

 a given number) can be obtained by forming a running sum of the frequencies (last column 

 of Table 14). 



Table 14. Probability density function for z = x + y. 



n 



Probability 



Frequency 



Cumulative frequency 







P<n) 



P(n) 



2 



1/36 



1/36 



1/36 



3 



(1 + l)/36 



2/36 



3/36 



4 



(1 + 1 + l)/36 



3/36 



6/36 



5 



(1 + 1 + 1 + l)/36 



4/36 



10/36 



6 



(1 + 1 + 1 + 1 + l)/36 



5/36 



15/36 



7 



(1 + 1 + 1 + 1 + 1 + l)/36 



6/36 



20/36 



8 



(1 + 1 + 1 + 1 + l)/36 



5/36 



26/36 



9 



(1 + 1 + 1 + l)/36 



4/36 



30/36 



10 



(1 + 1 + l)/36 



3/36 



33/36 



11 



(1 + l)/36 



2/36 



35/36 



12 



1/36 



1/36 



36/36 



In most practical cases of importance, the probabilities of the individual elements of the 

 array are not uniform and the underlying principle is not easily seen. 



