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Sagot :
First, let us examine the data provided in the table to understand the problem. We have the following purchases for one month at an ice cream stand:
[tex]\[ \begin{array}{|l|c|c|c|} \hline & \text{Smoothie} & \text{Shake} & \text{Ice Cream} \\ \hline \text{Strawberry} & 41 & 53 & 43 \\ \hline \text{Apple} & 73 & 59 & 37 \\ \hline \text{Banana} & 89 & 13 & 29 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
#### Step 1: Calculate the Total Number of Entries
To find the total number of customers who made a purchase, we sum all the values in the table:
[tex]\[ 41 + 73 + 89 + 53 + 59 + 13 + 43 + 37 + 29 = 437 \][/tex]
So, the total number of entries is [tex]\( 437 \)[/tex].
#### Step 2: Calculate the Number of Customers Who Bought a Shake
We add the values under the "Shake" column:
[tex]\[ 53 \, (\text{Strawberry}) + 59 \, (\text{Apple}) + 13 \, (\text{Banana}) = 125 \][/tex]
So, the number of customers who bought a shake is [tex]\( 125 \)[/tex].
#### Step 3: Calculate the Number of Customers Who Bought Strawberry Items
We add the values under the "Strawberry" row:
[tex]\[ 41 \, (\text{Smoothie}) + 53 \, (\text{Shake}) + 43 \, (\text{Ice Cream}) = 137 \][/tex]
So, the number of customers who bought strawberry items is [tex]\( 137 \)[/tex].
#### Step 4: Avoid Double-Counting Customers Who Bought Both Strawberry and Shake
To avoid double-counting, we identify the customers who bought a strawberry shake.
[tex]\[ 53 \][/tex]
So, the number of customers who bought both strawberry and shake is [tex]\( 53 \)[/tex].
#### Step 5: Calculate the Number of Customers Who Bought a Shake or It is Strawberry
We use the principle of inclusion and exclusion to avoid double-counting:
[tex]\[ \text{Shake or Strawberry} = \text{Shake Customers} + \text{Strawberry Customers} - \text{Customers who bought both Shake and Strawberry} \][/tex]
[tex]\[ \text{Shake or Strawberry} = 125 + 137 - 53 = 209 \][/tex]
#### Step 6: Calculate the Probability in Simplest Form
The probability is given by the ratio of the number of customers who bought a shake or it is strawberry to the total number of customers:
[tex]\[ P(\text{Shake or Strawberry}) = \frac{209}{437} \][/tex]
#### Step 7: Express the Probability as a Decimal
This fraction simplifies to approximately:
[tex]\[ \frac{209}{437} \approx 0.47826 \][/tex]
Thus, the probability that a randomly chosen customer has purchased a shake or it is strawberry is approximately [tex]\(0.47826\)[/tex] or 47.826%.
So, the final answer is:
[tex]\[ P (\text{Strawberry or Shake}) = \frac{209}{437} \][/tex]
[tex]\[ \begin{array}{|l|c|c|c|} \hline & \text{Smoothie} & \text{Shake} & \text{Ice Cream} \\ \hline \text{Strawberry} & 41 & 53 & 43 \\ \hline \text{Apple} & 73 & 59 & 37 \\ \hline \text{Banana} & 89 & 13 & 29 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
#### Step 1: Calculate the Total Number of Entries
To find the total number of customers who made a purchase, we sum all the values in the table:
[tex]\[ 41 + 73 + 89 + 53 + 59 + 13 + 43 + 37 + 29 = 437 \][/tex]
So, the total number of entries is [tex]\( 437 \)[/tex].
#### Step 2: Calculate the Number of Customers Who Bought a Shake
We add the values under the "Shake" column:
[tex]\[ 53 \, (\text{Strawberry}) + 59 \, (\text{Apple}) + 13 \, (\text{Banana}) = 125 \][/tex]
So, the number of customers who bought a shake is [tex]\( 125 \)[/tex].
#### Step 3: Calculate the Number of Customers Who Bought Strawberry Items
We add the values under the "Strawberry" row:
[tex]\[ 41 \, (\text{Smoothie}) + 53 \, (\text{Shake}) + 43 \, (\text{Ice Cream}) = 137 \][/tex]
So, the number of customers who bought strawberry items is [tex]\( 137 \)[/tex].
#### Step 4: Avoid Double-Counting Customers Who Bought Both Strawberry and Shake
To avoid double-counting, we identify the customers who bought a strawberry shake.
[tex]\[ 53 \][/tex]
So, the number of customers who bought both strawberry and shake is [tex]\( 53 \)[/tex].
#### Step 5: Calculate the Number of Customers Who Bought a Shake or It is Strawberry
We use the principle of inclusion and exclusion to avoid double-counting:
[tex]\[ \text{Shake or Strawberry} = \text{Shake Customers} + \text{Strawberry Customers} - \text{Customers who bought both Shake and Strawberry} \][/tex]
[tex]\[ \text{Shake or Strawberry} = 125 + 137 - 53 = 209 \][/tex]
#### Step 6: Calculate the Probability in Simplest Form
The probability is given by the ratio of the number of customers who bought a shake or it is strawberry to the total number of customers:
[tex]\[ P(\text{Shake or Strawberry}) = \frac{209}{437} \][/tex]
#### Step 7: Express the Probability as a Decimal
This fraction simplifies to approximately:
[tex]\[ \frac{209}{437} \approx 0.47826 \][/tex]
Thus, the probability that a randomly chosen customer has purchased a shake or it is strawberry is approximately [tex]\(0.47826\)[/tex] or 47.826%.
So, the final answer is:
[tex]\[ P (\text{Strawberry or Shake}) = \frac{209}{437} \][/tex]
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