To find the probability that a randomly chosen customer either purchased a shake or purchased a strawberry item, we need to start by determining the relevant totals.
1. Calculate the total number of purchases:
- Strawberry Smoothie: 41
- Strawberry Shake: 53
- Strawberry Ice Cream: 43
- Apple Smoothie: 73
- Apple Shake: 59
- Apple Ice Cream: 37
- Banana Smoothie: 89
- Banana Shake: 13
- Banana Ice Cream: 29
Adding these together gives us the total number of purchases:
[tex]\[
41 + 53 + 43 + 73 + 59 + 37 + 89 + 13 + 29 = 437
\][/tex]
So, the total number of purchases is 437.
2. Calculate the number of strawberry purchases:
- Strawberry Smoothie: 41
- Strawberry Shake: 53
- Strawberry Ice Cream: 43
Adding these together gives us:
[tex]\[
41 + 53 + 43 = 137
\][/tex]
So, the total number of strawberry purchases is 137.
3. Calculate the number of shake purchases:
- Strawberry Shake: 53
- Apple Shake: 59
- Banana Shake: 13
Adding these together gives us:
[tex]\[
53 + 59 + 13 = 125
\][/tex]
So, the total number of shake purchases is 125.
4. Determine the overlap (strawberry shakes):
- As strawberry shakes are included in both the strawberry and shake categories, we must subtract them once to prevent double-counting.
- Number of Strawberry Shakes: 53
5. Calculate the total number of purchases that are either shakes or strawberry:
[tex]\[
137 + 125 - 53 = 209
\][/tex]
6. Calculate the probability:
The probability [tex]\( P \)[/tex] of choosing a customer who purchased either a shake or a strawberry item is given by the ratio of the number of desired outcomes to the total number of outcomes.
[tex]\[
P(\text{Strawberry or Shake}) = \frac{209}{437}
\][/tex]
7. Simplify the fraction:
Find the greatest common divisor (GCD) of the numerator (209) and the denominator (437). The GCD is 19.
So, we simplify the fraction by dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{209 \div 19}{437 \div 19} = \frac{11}{23}
\][/tex]
Therefore, the probability that a randomly chosen customer either purchased a shake or a strawberry item, in the simplest form, is:
[tex]\[
P(\text{Strawberry or Shake}) = \frac{11}{23}
\][/tex]
