ive Literacy Hannah Thompson 03/20/259:19 PM Question 7, 11.4.18 HW Score: 59.33\%, 5.83 of 10 points Patiocto Points: 0 of 1 (e) below. Click here fo viow tre dogran B Dotermine the saingie specce. A. WYN.YANY,NN,NAAYANAA o. YN,YANY,NAAYAN c) Determine the protsatity thet a board member votes abstain, no, yes in that order of the motions. \( \frac{1}{27} \) (Type an intager or a elimplited frection.) d) Determine the probability that a board menter votes yes on exaclly one of the motions. \( \frac{4}{9} \) (Type an intager or a simpeled fraction) e) Detemine the probabily that a board member voles yes on at leest one motion. \( \square \) (Type an integer or a oimpliled fraction)

The voting behavior for each motion is modeled such that a board member can vote: - Yes (Y) - No (N) - Abstain (A) Since each vote is independent, the total number of outcomes (the sampling space) for three motions is \[ 3^3 = 27. \] For each motion, the probability of any specified vote (Y, N, or A) is \[ \frac{1}{3}. \] **(c) Probability of voting abstain, no, yes in that order** The probability is calculated by multiplying the independent probabilities: \[ P(A, N, Y) = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}. \] **(d) Probability of voting yes on exactly one of the motions** There are \(\binom{3}{1} = 3\) ways to choose which motion gets a yes. For the chosen motion, the probability of yes is \(\frac{1}{3}\), and for each of the other two motions, the probability of not yes (either no or abstain) is \(\frac{2}{3}\). Thus: \[ P(\text{exactly one yes}) = \binom{3}{1} \cdot \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = 3 \cdot \frac{1}{3} \cdot \frac{4}{9} = \frac{4}{9}. \] **(e) Probability of voting yes on at least one motion** The complement is voting no yes on any motion (i.e., voting not yes on all three motions). The probability for not yes on any single motion is \(\frac{2}{3}\), so: \[ P(\text{no yes votes}) = \left(\frac{2}{3}\right)^3 = \frac{8}{27}. \] Thus, the probability of voting yes on at least one motion is: \[ P(\text{at least one yes}) = 1 - \frac{8}{27} = \frac{19}{27}. \]