The Orange County Fair offers raffle tickets to its guests. Three guests will win $\$ 500$ prizes, six guests will win $\$ 110$ prizes, and thirteen guests will win $\$ 16$ prizes. There were 60 tickets sold for $\$ 8$ each.
a) Construct a probability distribution for the data.
\begin{tabular}{|c|c|}
\hline $\begin{array}{c}\text { Select an answer } 0 \\
\text { (No commas and from least } \\
\text { to greatest) }\end{array}$ \\
Select an answer 0 \\
(No commas) & \\
\hline $\begin{array}{c}\text { Select an answer } 0 \\
\text { (Write as a decimal rounded } \\
\text { to two decimal places) }\end{array}$ & \\
\hline
\end{tabular}

Step 1 ：The Orange County Fair offers raffle tickets to its guests. Three guests will win $500 prizes, six guests will win $110 prizes, and thirteen guests will win $16 prizes. There were 60 tickets sold for $8 each.

Step 2 ：To construct a probability distribution, we need to calculate the probability of each event. In this case, the events are winning a $500 prize, a $110 prize, a $16 prize, or no prize. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of outcomes. Here, the total number of outcomes is the total number of tickets sold, which is 60.

Step 3 ：The total number of tickets is 60. The prizes are $500, $110, $16, and $0. The number of winners for each prize are 3, 6, 13, and 38 respectively.

Step 4 ：The probabilities for each prize are calculated as follows: \(\frac{3}{60} = 0.05\) for the $500 prize, \(\frac{6}{60} = 0.1\) for the $110 prize, \(\frac{13}{60} = 0.2167\) for the $16 prize, and \(\frac{38}{60} = 0.6333\) for no prize.

Step 5 ：\(\boxed{\text{Final Answer: The probability distribution for the data is as follows:}}\)

Step 6 ：\begin{tabular}{|c|c|}

Step 7 ：\hline

Step 8 ：Prize & Probability \\

Step 9 ：\hline

Step 10 ：\$500 & 0.05 \\

Step 11 ：\$110 & 0.1 \\

Step 12 ：\$16 & 0.2167 \\

Step 13 ：\$0 & 0.6333 \\

Step 14 ：\hline

Step 15 ：\end{tabular}