The owner of a local movie theater keeps track of the number of tickets sold in each purchase. The owner determines the probabilities based on these records. Let [tex]$X$[/tex] represent the number of tickets bought in one purchase. The distribution for [tex]$X$[/tex] is given in the table.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Number of Tickets & 1 & 2 & 3 & 4 & 5 \\
\hline
Probability & 0.17 & 0.55 & 0.20 & 0.06 & 0.02 \\
\hline
\end{tabular}

Which of the following correctly represents the probability of a randomly selected purchase having at most three tickets?

A. [tex]$P(X \leq 3)$[/tex]
B. [tex][tex]$P(X\ \textless \ 3)$[/tex][/tex]
C. [tex]$P(X \geq 3)$[/tex]
D. [tex]$P(X\ \textgreater \ 3)$[/tex]



Answer :

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].