titative Literacy Hannah thompson 03/20/25 9:18 awork Question 7, 11,4.18 Part 5 of 8 HW Score: $$ 58.33 \%, 6.83 $$ of 10 points Points: 0 of 1 At a homeowners' association meeting, a board member can vote yes, vote no, or abstain on a motion. There are three motions on which each board member must vote. Complete parts (e) below. Click here to view tree diagram D Click here to view tree diagram C Click here to view tree diagram À Click here to view tree diagram B Determine the sample space. A. $$ Y Y, Y N, Y A, N Y, N N, N A, A Y, A N, A A $$ B. $$ M Y, Y N, Y Y A, Y N Y, Y N N, Y N A, Y A Y, Y A N, Y A A, N Y Y, N Y N, N Y A, N N Y, N N N, N N A, N A Y, N A N, N A A, A Y Y, A Y N, A Y A, A N Y, A N N, A N A, A A Y, A A N, A A A $$ C. $$ \begin{array}{l} \ Y\end{array}, Y Y, Y Y A, N N A, Y N N, Y N A, Y A Y, N A N, Y A A, N Y Y, N Y N, N A A, N N Y, N N N, A Y Y, A Y A, A N N, A N A, A A Y, A A A $$ D. YN,YA,NY,NA,AY,AN c) Determine the probability that a board member votes abstain, no, yes in that order of the motions. $$ \square $$ $$ \frac{1}{27} $$ (Type an integet or a simplified fraction.) d) Determine the probability that a board member votes yes on exactly one of the motions. $$ \square $$ (Type an integer or a simplified fraction.)

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UpStudy AI · Accepted Answer

**Step 1. Determine the total number of outcomes in the sample space.**

Each board member votes on 3 motions, and for each motion there are 3 choices: yes, no, or abstain. Therefore, the total number of outcomes is
$$
3^3 = 27.
$$

**Step 2. Part (c): Probability of voting abstain, no, yes in order.**

For the first motion, the vote must be abstain; for the second, no; and for the third, yes. There is exactly 1 outcome that matches this specific sequence.

Thus, the probability is 
$$
\frac{1}{27}.
$$

**Step 3. Part (d): Probability of voting yes on exactly one motion.**

- First, choose which one of the 3 motions will have a yes vote. There are 
$$
\binom{3}{1} = 3 \text{ ways.}
$$
- For the remaining 2 motions, each vote can be either no or abstain (2 options for each motion). Therefore, there are 
$$
2 \times 2 = 4 \text{ outcomes for the other motions.}
$$
- Thus, the total number of outcomes with exactly one yes vote is 
$$
3 \times 4 = 12.
$$

The probability of exactly one yes vote is then
$$
\frac{12}{27} = \frac{4}{9}.
$$

**Final Answers:**

c) $$\frac{1}{27}$$

d) $$\frac{4}{9}$$
**Sample Space:** A. $$ Y Y, Y N, Y A, N Y, N N, N A, A Y, A N, A A $$ **Probability of Voting Abstain, No, Yes in Order:** $$\frac{1}{27}$$ **Probability of Voting Yes on Exactly One Motion:** $$\frac{4}{9}$$

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