At a​ homeowners' association​ meeting, a board member can vote yes or on a motion.

There are three motions on which each board member must vote. Complete parts ​(​a) through ​(c) below.
​(a) Determine the number of points in the sample space.

​(b) Determine the probability that a board member votes yes on exactly two of the motions.

​(c) Determine the probability that a board member votes yes on at least one motion.

To answer these questions, let's break them down step by step:

(a) Determine the number of points in the sample space.
The sample space represents all the possible outcomes of the board member's vote on the three motions. Since each board member can either vote yes or no on each motion, there are two possible outcomes for each motion. Therefore, the total number of points in the sample space can be calculated as 2 * 2 * 2 = 8.

(b) Determine the probability that a board member votes yes on exactly two of the motions.
To calculate this probability, we need to know the number of favorable outcomes (outcomes where the board member votes yes on exactly two motions) and divide it by the total number of possible outcomes (the sample space).

If we choose two motions out of the three where the board member votes yes, there are 3C2 = 3 possible combinations. For each combination, the remaining motion will have the board member voting no. Therefore, the number of favorable outcomes is 3.

The total number of possible outcomes, as we calculated in part (a), is 8.

So, the probability that a board member votes yes on exactly two of the motions is 3/8.

(c) Determine the probability that a board member votes yes on at least one motion.
To calculate this probability, we need to find the number of favorable outcomes where the board member votes yes on at least one motion.

Considering the complement rule, we can find the number of favorable outcomes where the board member votes no on all three motions. Since there are only two possible outcomes (yes or no) for each motion, then the number of favorable outcomes where the board member votes no on all three motions is simply 1.

Therefore, the number of favorable outcomes where the board member votes yes on at least one motion is the complement of the above: 8 - 1 = 7.

The total number of possible outcomes, as calculated in part (a), is 8.

So, the probability that a board member votes yes on at least one motion is 7/8.