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 african americans, 2,745 native hawaiians, 12,831 latinos, 8,378 japanese americans, and 7,650 whites. of the people smoking 11 to 20 cigarettes per day, there were 6,514 african americans, 3,062 native hawaiians, 4,932 latinos, 10,680 japanese americans, and 9,877 whites. of the people smoking 21 to 30 cigarettes per day, there were 1,671 african americans, 1,419 native hawaiians, 1,406 latinos, 4,715 japanese americans, and 6,062 whites. of the people smoking at least 31 cigarettes per day, there were 759 african americans, 788 native hawaiians, 800 latinos, 2,305 japanese americans, and 3,970 whites.
in words, explain what it means to pick one person from the study who is \japanese american and smokes 21 to 30 cigarettes per day.\
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: given that the person is japanese american, the person smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person does not meet either criteria: neither japanese american nor smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: given that the person smokes 21 to 30 cigarettes, the person is japanese american.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both japanese american and smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet at least one of the criteria: either japanese american or smokes 21 to 30 cigarettes.
find the probability. (enter your probability as a fraction.)

Question

Accepted Answer

### Part 1: Interpreting "Japanese American AND smokes 21 to 30 cigarettes per day"
- The word "AND" in probability (and set theory) means that both conditions must be satisfied. So a person who is "Japanese American AND smokes 21 to 30 cigarettes per day" must be Japanese American **and** also smoke 21 - 30 cigarettes per day.
- Let's analyze the options:
    - Option 1: Talks about "given that the person is Japanese American", which is a conditional probability (using "given"), but we need a joint event (both conditions at the same time), so this is incorrect.
    - Option 2: Says "does not meet either criteria", which is the opposite of "AND" (that would be "NEITHER...NOR"), so this is incorrect.
    - Option 3: Talks about "given that the person smokes 21 to 30 cigarettes", which is a conditional probability, not a joint event, so this is incorrect.
    - Option 4: Says "the person has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes", which correctly defines the "AND" event (both conditions are satisfied simultaneously).

### Part 2: Calculating the Probability
To find the probability, we use the formula for probability of an event $ E $:
$ P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} $

#### Step 1: Find the number of favorable outcomes (Japanese American and smokes 21 - 30 cigarettes per day)
From the data, the number of Japanese Americans who smoke 21 - 30 cigarettes per day is 4,715.

#### Step 2: Find the total number of people in the study
We need to sum the number of people in each ethnic group and each smoking category:

- **At most 10 cigarettes per day**:
$ 9,886 + 2,745 + 12,831 + 8,378 + 7,650 = 9,886+2,745 = 12,631; 12,631+12,831 = 25,462; 25,462+8,378 = 33,840; 33,840+7,650 = 41,490 $

- **11 - 20 cigarettes per day**:
$ 6,514 + 3,062 + 4,932 + 10,680 + 9,877 = 6,514+3,062 = 9,576; 9,576+4,932 = 14,508; 14,508+10,680 = 25,188; 25,188+9,877 = 35,065 $

- **21 - 30 cigarettes per day**:
$ 1,671 + 1,419 + 1,406 + 4,715 + 6,062 = 1,671+1,419 = 3,090; 3,090+1,406 = 4,496; 4,496+4,715 = 9,211; 9,211+6,062 = 15,273 $

- **At least 31 cigarettes per day**:
$ 759 + 788 + 800 + 2,305 + 3,970 = 759+788 = 1,547; 1,547+800 = 2,347; 2,347+2,305 = 4,652; 4,652+3,970 = 8,622 $

Now, sum all these totals:
$ 41,490 + 35,065 + 15,273 + 8,622 $

First, $ 41,490 + 35,065 = 76,555 $
Then, $ 76,555 + 15,273 = 91,828 $
Then, $ 91,828 + 8,622 = 100,450 $

#### Step 3: Calculate the probability
The number of favorable outcomes (Japanese American, 21 - 30 cigarettes) is 4,715.
Total number of outcomes is 100,450.

So the probability $ P = \frac{4,715}{100,450} $. We can simplify this fraction by dividing numerator and denominator by 5:
$ \frac{4,715 \div 5}{100,450 \div 5} = \frac{943}{20,090} $

### Final Answers
- Interpretation: The correct option is **Option 4** (To pick one person from the study who is Japanese American and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes).
- Probability: $ \boldsymbol{\frac{943}{20090}} $ (or $ \frac{4715}{100450} $ before simplification)

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QUESTION IMAGE

Question

Explanation:

Part 1: Interpreting "Japanese American AND smokes 21 to 30 cigarettes per day"

Part 2: Calculating the Probability

Step 1: Find the number of favorable outcomes (Japanese American and smokes 21 - 30 cigarettes per day)

Step 2: Find the total number of people in the study

Step 3: Calculate the probability

Final Answers

Answer: