Chapter 5: Q 97. (page 331)

Lactose intolerance Lactose intolerance causes difficulty
in digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States
(ignoring other groups and people who consider themselves to belong to more than one race), $82 %$ of the population is white, 14% is black, and $4 %$ is
Asian. Moreover, $15 %$ of whites, $70 %$ of blacks, and $90 %$ of Asians are lactose intolerant. $20$
(a) What percent of the entire population is lactose intolerant?
(b) What percent of people who are lactose intolerant are Asian?

Expert verified

Part (a) The percentage of the entire population of being lactose intolerant is $25.70 %$

Part (b) $P (A s i a n | l a c t o s e) = 14.01 %$

Step by step solution

P (W h i t e) = 82 % = 0.82 P (B l a c k) = 14 % = 0.14 P (A s i a n) = 4 % = 0.04

Multiplication rule: $P (A \cap B \cap C) = P (A) \times P (B) \times P (C)$

In response to the question:

P (l a c t o s e | w h i t e) = 15 % = 0.15 P (l a c t o s e | b l a c k) = 70 % = 0.70 P (l a c t o s e | A s i a n) = 90 % = 0.90

Now, for each category, calculate the likelihood of being lactose intolerant:

P (l a c t o s e \cap w h i t e) = P (l a c t o s e | w h i t e) \times P (w h i t e) = 0.15 \times 0.82 = 0.2464 P (l a c t o s e \cap b l a c k) = P (l a c t o s e | b l a c k) \times P (b l a c k) = 0.70 \times 0.14 = 0.098 P (l a c t o s e | \cap A s i a n) = P (l a c t o s e | A s i a n) \times P (A s i a n) = 0.90 \times 0.04 = 0.0360

Fill in the blanks with the corresponding probabilities:

P (l a c t o s e) = 0.1230 + 0.0980 + 0.360 = 0.2570 \Rightarrow P (l a c t o s e) = 0.2570 \times 100 = 25.70 %

As a result, $25.70$ percent of the population suffers from lactose intolerance.

Conditional probability: $P (A | B) = P (A \cap B) / P (A)$

$P (l a c t o s e) = 0.2570$

Conditional probability: $P (A | B) = P (A \cap B) / P (A)$

As a result, we have:

$P (A s i a n | l a c t o s e) = \frac{P (l a c t o s e \cap A s i a n)}{P (A s i a n)} = \frac{0.360}{0.2570} \approx 0.1401 = 14.01 %$

As a result, Asians account for 14.01 percent of lactose-intolerant persons.

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