Question

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

In words, explain what it means to pick one person from the study who is “Japanese American or smokes $21to30$cigarettes per day.” Also, find the probability.

Short answer

In this question "OR" represents the union, which means we must select a person who is either Japanese American or who smokes $21to30$cigarettes per day.

The required solution is $0.364.$

Step-by-step solution

Given information

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

Calculation

We know, according to the table,

In the given question "OR" represents the union, which means we must select a person who is either Japanese American or who smokes $21to30$cigarettes per day.

To obtain the probability we use the formula $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B).$

$$\begin{gathered} P\left(A\right)=\frac{26078}{100450} \\ P\left(B\right)=\frac{15273}{100450} \\ P(A\cap B)=\frac{4715}{100450} \\ P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B) \\ P(A\cup B)=\frac{26078}{100450}+\frac{15273}{100450}-\frac{4715}{100450} \\ P(A\cup B)=0.364 \end{gathered}$$

Final answer

In this question "OR" represents the union, which means we must select a person who is either Japanese American or who smokes $21to30$cigarettes a day.

The required solution is$0.364.$