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At an ice cream stand, the purchases for one month are recorded in the table below:

\begin{tabular}{|l|c|c|c|}
\hline & Smoothie & Shake & Ice Cream \\
\hline Strawberry & 41 & 53 & 43 \\
\hline Apple & 73 & 59 & 37 \\
\hline Banana & 89 & 13 & 29 \\
\hline
\end{tabular}

If we choose a customer at random, what is the probability that they have purchased a banana smoothie?

[tex]\[
P (\text{Banana and Smoothie}) = \frac{89}{?}
\][/tex]

Give your answer in simplest form.



Answer :

GinnyAnswer
First, we need to determine the total number of purchases made at the ice cream stand for the month. We do this by summing the number of purchases in each category (Smoothie, Shake, Ice Cream) across all flavors (Strawberry, Apple, Banana).

Here is the breakdown of the calculation:

1. For Strawberry:
- Smoothies: 41
- Shakes: 53
- Ice Cream: 43
[tex]\[ \text{Total for Strawberry} = 41 + 53 + 43 \][/tex]

2. For Apple:
- Smoothies: 73
- Shakes: 59
- Ice Cream: 37
[tex]\[ \text{Total for Apple} = 73 + 59 + 37 \][/tex]

3. For Banana:
- Smoothies: 89
- Shakes: 13
- Ice Cream: 29
[tex]\[ \text{Total for Banana} = 89 + 13 + 29 \][/tex]

Next, we add these totals together to find the overall total number of purchases:

[tex]\[ \text{Total number of purchases} = (41 + 53 + 43) + (73 + 59 + 37) + (89 + 13 + 29) \][/tex]

This sum is:
[tex]\[ 137 + 169 + 131 = 437 \][/tex]

So, the total number of purchases is 437.

Next, we need to identify how many of these purchases were for banana smoothies. From the table, we see that the number of banana smoothie purchases is 89.

Therefore, the probability [tex]\( P(\text{Banana and Smoothie}) \)[/tex] that a randomly chosen customer purchased a banana smoothie is:

[tex]\[ P(\text{Banana and Smoothie}) = \frac{\text{Number of banana smoothie purchases}}{\text{Total number of purchases}} = \frac{89}{437} \][/tex]

When expressed as a fraction and having verified the calculations, the probability simplifies to approximately 0.2037. Hence, the probability that a randomly chosen customer purchased a banana smoothie is:

[tex]\[ P(\text{Banana and Smoothie}) = \frac{89}{437} \][/tex]
kenneth7751

Answer:

Step-by-step explanation:

To determine the probability that a randomly chosen customer purchased a banana smoothie, we first need to find the total number of purchases of banana smoothies and then divide it by the total number of all purchases.

From the table provided:

- Number of banana smoothies purchased: 89

To find the total number of all purchases, sum up all the entries in the table:

\[

41 + 53 + 43 + 73 + 59 + 37 + 89 + 13 + 29 = 437

\]

Now, the probability \( P \) that a randomly chosen customer purchased a banana smoothie is calculated as:

\[

P = \frac{\text{Number of banana smoothies}}{\text{Total number of purchases}} = \frac{89}{437}

\]

To simplify this fraction, we find the greatest common divisor (GCD) of 89 and 437, which is 1 (since 89 is a prime number). Therefore, the simplified probability is:

\[

P = \frac{89}{437}

\]

Thus, the probability that a randomly chosen customer purchased a banana smoothie is \( \boxed{\frac{89}{437}} \).

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