Question 1044884
<font color=black size=3>
Y = member votes yes
N = member votes no
A = member abstains


Sample Space
YYY
YYN
YYA
YNY
YNN
YNA
YAY
YAN
YAA


NYY
NYN
NYA
NNY
NNN
NNA
NAY
NAN
NAA


AYY
AYN
AYA
ANY
ANN
ANA
AAY
AAN
AAA


Those are all the possible outcomes. There are 27 items in the sample space above. There's only one case of YYY out of the 27 cases total. So the probability of voting all yes is 1/27.



There are 8 cases where Y doesn't show up at all. Those cases are

NNN
NNA
NAN
NAA

ANN
ANA
AAN
AAA


This means there are 27-8 = 19 cases where Y shows up at least once. 


So the probability of voting at least one yes (Y) is 19/27



-------------------------------------------------------------------------

Summary:


The answer to the problem <font color=blue>"Determine the number of points in a sample space."</font> is <font color=red>27</font> 


The answer to the problem <font color=blue>"Determine the probability that a board member votes yes on all of the motions."</font> is the fraction <font color=red>1/27</font> 


The answer to the problem <font color=blue>"Determine the probability that a board member votes yes on at least one."</font> is the fraction <font color=red>19/27</font> 


</font>