At a concession stand, the combo purchases for one week are recorded in the table below:

\begin{tabular}{|l|c|c|c|}
\hline & Hamburger & Pizza & \begin{tabular}{c}
Deli \\
Sandwich
\end{tabular} \\
\hline French Fries & 83 & 67 & 37 \\
\hline Peanuts & 2 & 5 & 14 \\
\hline Popcorn & 19 & 29 & 3 \\
\hline
\end{tabular}

If we choose a customer at random, what is the probability that they have purchased French fries and a deli sandwich?

[tex]\[ P(\text{French Fries and Deli Sandwich}) = \][/tex]

Give your answer in simplest form.



Answer :

To find the probability that a randomly chosen customer purchased french fries and a deli sandwich, we need to identify the relevant numbers and then use the probability formula.

1. Identify total number of combo purchases:
We sum up all the purchases from the entire table, which includes hamburgers, pizzas, and deli sandwiches, combined with french fries, peanuts, and popcorn.

[tex]\[ \begin{align*} \text{Total purchases} & = \text{French Fries with Hamburger} + \text{French Fries with Pizza} + \text{French Fries with Deli Sandwich} \\ & + \text{Peanuts with Hamburger} + \text{Peanuts with Pizza} + \text{Peanuts with Deli Sandwich} \\ & + \text{Popcorn with Hamburger} + \text{Popcorn with Pizza} + \text{Popcorn with Deli Sandwich} \\ & = 83 + 67 + 37 \\ & + 2 + 5 + 14 \\ & + 19 + 29 + 3 \\ & = 259 \end{align*} \][/tex]

2. Identify the number of times french fries and a deli sandwich were purchased together:

The given value for french fries and deli sandwich purchases is [tex]\( 37 \)[/tex].

3. Calculate the probability:

The probability [tex]\( P \)[/tex] can be calculated using the ratio of the number of favorable outcomes (purchases of french fries and deli sandwiches) to the total number of outcomes (all purchases).

[tex]\[ P(\text{French Fries and Deli Sandwich}) = \frac{\text{Number of French Fries and Deli Sandwich purchases}}{\text{Total number of purchases}} \][/tex]

Substituting the values:

[tex]\[ P(\text{French Fries and Deli Sandwich}) = \frac{37}{259} \][/tex]

This fraction is already in its simplest form.

Hence, the probability that a randomly chosen customer purchased french fries and a deli sandwich is:

[tex]\[ \boxed{\frac{37}{259}} \][/tex]

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