To determine the correct representation of the probability of a randomly selected purchase having at most three tickets, we need to understand what "at most three tickets" means.
"At most three tickets" means the number of tickets bought can be 1, 2, or 3. Therefore, the correct probability expression we are looking for is [tex]\(P(X \leq 3)\)[/tex], which denotes the probability that the number of tickets [tex]\(X\)[/tex] is less than or equal to 3.
Given the distribution of [tex]\(X\)[/tex]:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline \begin{tabular}{c}
Number of \\
Tickets
\end{tabular} & 1 & 2 & 3 & 4 & 5 \\
\hline Probability & 0.17 & 0.55 & 0.20 & 0.06 & 0.02 \\
\hline
\end{tabular}
\][/tex]
To find [tex]\(P(X \leq 3)\)[/tex], we sum the probabilities of [tex]\(X\)[/tex] being 1, 2, or 3.
[tex]\[
P(X \leq 3) = P(X = 1) + P(X = 2) + P(X = 3)
\][/tex]
From the table, we have:
[tex]\[
P(X = 1) = 0.17, \quad P(X = 2) = 0.55, \quad P(X = 3) = 0.20
\][/tex]
Now, we add these probabilities:
[tex]\[
P(X \leq 3) = 0.17 + 0.55 + 0.20
\][/tex]
Calculating this sum:
[tex]\[
P(X \leq 3) = 0.92
\][/tex]
Thus, the probability that a randomly selected purchase has at most three tickets is [tex]\(0.92\)[/tex]. Therefore, the correct representation is [tex]\(P(X \leq 3)\)[/tex].
