Chapter 5: Q 84. (page 330)

Urban voters The voters in a large city are $40 %$ white, $40 %$ black, and $20 %$ Hispanic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks.) A mayoral candidate anticipates attracting $30 %$ of the white vote, $90 %$ of the black vote, and $50 %$ of the Hispanic vote. Draw a tree diagram to represent this situation. What percent of the overall vote does the candidate expect to get? Use the four-step process to guide your work.

Expert verified

The probability of support for each political book is $58 %$

Step by step solution

In a major city, $40$ percent of white voters, $40$ percent black voters, and $20 %$ Hispanic voters make up the voting population.

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

The tree diagram is shown below, according to the question:

Now we must determine the likelihood of support for each political book.

P (w h i t e \cap S u p p o r t) = P (S u p p o r t / w h i t e) \times P (w h i t e) = 40 % \times 30 % = 0.40 \times 0.30 = 0.12

P (B l a c k \cap S u p p o r t) = P (S u p p o r t / B l a c k) \times P (B l a c k) = 40 % \times 30 % = 0.40 \times 0.90 = 0.36

$P (H i s p a n i c \cap S u p p o r t) = P (S u p p o r t / H i s p a n i c) \times P (H i s p a n i c) = 20 % \times 50 %$

Fill in the blanks with the corresponding probabilities:

P (S u p p o r t) = 0.1200 + 0.3600 + 0.1000 = 0.5800 = 58 %

As a result, any political book has a $58 %$ chance of being supported.

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