Questions: At an ice cream stand, the purchases for one month are recorded in the table below: Smoothie Shake Ice Cream ------------------------------------------ Strawberry 41 53 43 Apple 73 59 37 Banana 89 13 29 If we choose a customer at random, what is the probability that they have purchased a shake or it is strawberry? P( Strawberry or Shake )=

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

Smoothie Shake Ice Cream
------------------------------------------
Strawberry 41 53 43
Apple 73 59 37
Banana 89 13 29

If we choose a customer at random, what is the probability that they have purchased a shake or it is strawberry?
P( Strawberry or Shake )=

Transcript text: 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 shake or it is strawberry? $P($ Strawberry or Shake $)=$ $\square$ Give your answer in simplest form.

Solution

Solution Steps

To find the probability that a randomly chosen customer purchased a shake or it is strawberry, we need to use the principle of inclusion-exclusion. First, calculate the total number of purchases. Then, find the number of purchases that are either a shake or strawberry. Finally, divide the number of favorable outcomes by the total number of purchases to get the probability.

Step 1: Calculate Total Purchases

The total number of purchases is the sum of all the individual purchases for smoothies, shakes, and ice creams across all flavors. This is given by:

\[ \text{Total Purchases} = 41 + 53 + 43 + 73 + 59 + 37 + 89 + 13 + 29 = 437 \]

Step 2: Calculate Purchases of Shakes

The total number of shake purchases is the sum of shakes across all flavors:

\[ \text{Shake Purchases} = 53 + 59 + 13 = 125 \]

Step 3: Calculate Purchases of Strawberry

The total number of strawberry purchases is the sum of all strawberry items:

\[ \text{Strawberry Purchases} = 41 + 53 + 43 = 137 \]

Step 4: Apply Inclusion-Exclusion Principle

To find the number of purchases that are either a shake or strawberry, we use the inclusion-exclusion principle:

\[ \text{Shake or Strawberry} = \text{Shake Purchases} + \text{Strawberry Purchases} - \text{Strawberry Shakes} \]

\[ = 125 + 137 - 53 = 209 \]

Step 5: Calculate Probability

The probability that a randomly chosen customer purchased a shake or it is strawberry is given by the ratio of favorable outcomes to the total number of purchases:

\[ P(\text{Strawberry or Shake}) = \frac{\text{Shake or Strawberry}}{\text{Total Purchases}} = \frac{209}{437} \approx 0.4783 \]