$\text{An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were}9,886\text{African Americans,}2,745\text{Native Hawaiians,}12,831\text{Latinos,}8,378\text{Japanese Americans, and}7,650\text{Whites. Of the people smoking}11to20\text{cigarettes per day, there were}6,514\text{African Americans,}3,062\text{Native Hawaiians,}4,932\text{Latinos,}10,680\text{Japanese Americans, and}9,877\text{Whites. Of the people smoking}21to30\text{cigarettes per day, there were}1,671\text{African Americans,}1,419\text{Native Hawaiians,}1,406\text{Latinos,}4,715\text{Japanese Americans, and}6,062\text{Whites. Of the people smoking at least}31\text{cigarettes per day, there were}759\text{African Americans,}788\text{Native Hawaiians,}800\text{Latinos,}2,305\text{Japanese Americans, and}3,970\text{Whites}$

Prove that smoking level/day and ethnicity are dependent events.

$\text{An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were}9,886\text{African Americans,}2,745\text{Native Hawaiians,}12,831\text{Latinos,}8,378\text{Japanese Americans, and}7,650\text{Whites. Of the people smoking}11to20\text{cigarettes per day, there were}6,514\text{African Americans,}3,062\text{Native Hawaiians,}4,932\text{Latinos,}10,680\text{Japanese Americans, and}9,877\text{Whites. Of the people smoking}21to30\text{cigarettes per day, there were}1,671\text{African Americans,}1,419\text{Native Hawaiians,}1,406\text{Latinos,}4,715\text{Japanese Americans, and}6,062\text{Whites. Of the people smoking at least}31\text{cigarettes per day, there were}759\text{African Americans,}788\text{Native Hawaiians,}800\text{Latinos,}2,305\text{Japanese Americans, and}3,970\text{Whites}$

We know, according to the table,

E = event of ethnicity.

S = event for smoking level per day.

From the table, we can conclude that$P\left(E\right)isnotequaltoP(E/S).$

Thus, smocking level/day and ethnicity are not independent, which means they are dependent.

Hence, it is proved that smoking level/day and ethnicity are dependent events.

It is proved here that$P\left(E\right)isnotequaltoP(E/S).$