To determine the probability that a randomly chosen customer has purchased either a shake or a strawberry item, we can follow these steps methodically:
1. Collect the total number of purchases:
First, we need to sum all the purchases made:
[tex]\[
\text{Total purchases} = (41 + 73 + 89) + (53 + 59 + 13) + (43 + 37 + 29)
\][/tex]
Simplifying within each category:
[tex]\[
= 203 + 125 + 109 = 437
\][/tex]
Thus, the total number of purchases recorded is 437.
2. Calculate the number of strawberry purchases:
The count of strawberry purchases from all categories (Smoothie, Shake, Ice Cream):
[tex]\[
\text{Strawberry purchases} = 41 + 53 + 43 = 137
\][/tex]
3. Calculate the number of shake purchases:
The count of shake purchases from all flavors (Strawberry, Apple, Banana):
[tex]\[
\text{Shake purchases} = 53 + 59 + 13 = 125
\][/tex]
4. Account for the overlap of shake and strawberry purchases:
The count of purchases that are both shake and strawberry:
[tex]\[
\text{Both shake and strawberry purchases} = 53
\][/tex]
5. Use the principle of Inclusion-Exclusion to find the count of either strawberry or shake purchases:
The number of purchases that are either shakes or strawberry:
[tex]\[
\text{Either strawberry or shake} = \text{Strawberry purchases} + \text{Shake purchases} - \text{Both shake and strawberry purchases}
\][/tex]
Substituting the values we have:
[tex]\[
= 137 + 125 - 53 = 209
\][/tex]
6. Formulate the probability:
The probability of a randomly chosen customer having purchased either a shake or a strawberry item is the ratio of the number of favorable outcomes (either strawberry or shake) to the total number of outcomes (total purchases):
[tex]\[
P(\text{Strawberry or Shake}) = \frac{\text{Either strawberry or shake}}{\text{Total purchases}} = \frac{209}{437}
\][/tex]
So, the probability, expressed in simplest form, is:
[tex]\[
P(\text{Strawberry or Shake}) = \frac{209}{437}
\][/tex]
