Of people who died in the United States in a recent year, 86% were white, 12% were black, and 2% were Asian. (We will ignore the small number of deaths among other races.) Diabetes caused 2.8% of deaths among whites, 4.4% among blacks, and 3.5% among Asians. The probability that a randomly chosen death was due to diabetes is about

1. Let \( P(W) = 0.86 \), \( P(B) = 0.12 \), and \( P(A) = 0.02 \) be the probabilities of a death being white, black, or Asian, respectively.

2. Let \( P(D|W) = 0.028 \), \( P(D|B) = 0.044 \), and \( P(D|A) = 0.035 \) be the probabilities of a death being due to diabetes given the person is white, black, or Asian.

3. Use the law of total probability to find \( P(D) \):

P(D) = P(D|W) \cdot P(W) + P(D|B) \cdot P(B) + P(D|A) \cdot P(A)

4. Substitute the given probabilities:

P(D) = 0.028 \cdot 0.86 + 0.044 \cdot 0.12 + 0.035 \cdot 0.02

5. Calculate each term:

0.028 \cdot 0.86 = 0.02408

0.044 \cdot 0.12 = 0.00528

0.035 \cdot 0.02 = 0.0007

6. Sum the results:

P(D) = 0.02408 + 0.00528 + 0.0007 = 0.03006

7. Thus, the probability that a randomly chosen death was due to diabetes is:So, the probability that a randomly chosen death was due to diabetes is approximately 0.0301, or 3.01%.