Question

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.

Short answer

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

Step-by-step solution

Step 1. Given Information

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

Step 2. Concept Used

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

Step 3. Calculation

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

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

$P(white\cap Support)=P(Support/white)\times P\left(white\right)=40\%\times 30\%=0.40\times 0.30=0.12$$$\begin{gathered} P(Black\cap Support)=P(Support/Black)\times P\left(Black\right) \\ =40\%\times 30\% \\ =0.40\times 0.90 \\ =0.36 \end{gathered}$$

$$\begin{gathered} P(Hispanic\cap Support)=P(Support/Hispanic)\times P\left(Hispanic\right) \\ =20\%\times 50\% \end{gathered}$$

Fill in the blanks with the corresponding probabilities:

$$\begin{gathered} P\left(Support\right)=0.1200+0.3600+0.1000 \\ =0.5800 \\ =58\% \end{gathered}$$

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