A shoe company conducts a survey to determine the expected value of online sales for their new line of shoes coming out next fall. Based on past years, they have observed the following data on the probability of selling different types of shoes in the new line. The company expects that 1945 people will visit the website for their new line on launch day. Note that some of the online shoppers will not make a purchase.

sneakers $53.99 3/25 high heels $71.25 4/15 sandals $39.50 6/25 loafers $45.75 6/25


The shoe company can use the concept of **expected value** to predict the online sales from the 1,945 website visitors on launch day. Here’s how you calculate the expected value of sales:

1. **Identify the different categories** of shoes and their corresponding prices.
2. **Determine the probability** of selling each type of shoe (based on past years' data).
3. **Calculate the expected value** of the sales by multiplying each shoe type's price by its probability of being sold.

### Step 1: Define the probabilities and prices

From the problem, we know the following:

- **Sneakers**: Price = $53.99, Probability = $ \frac{3}{25} $
- **High heels**: Price = $71.25, Probability = $ \frac{4}{15} $
- **Sandals**: Price = $39.50, Probability = $ \frac{6}{25} $
- **Loafers**: Price = $45.75, Probability = $ \frac{6}{25} $

### Step 2: Compute the expected sales value per person

The expected value of sales for a single visitor is calculated as:

\[
E(\text{Sales}) = (\text{Price of Sneakers} \times \text{Probability of Sneakers}) + (\text{Price of High Heels} \times \text{Probability of High Heels}) + (\text{Price of Sandals} \times \text{Probability of Sandals}) + (\text{Price of Loafers} \times \text{Probability of Loafers})
\]

Substituting the given values:

\[
E(\text{Sales}) = (53.99 \times \frac{3}{25}) + (71.25 \times \frac{4}{15}) + (39.50 \times \frac{6}{25}) + (45.75 \times \frac{6}{25})
\]

### Step 3: Calculate the total expected sales

Next, you multiply the expected value of sales for one person by the total number of visitors (1,945) to estimate the total sales for launch day.

Let's calculate this now.The total expected sales for the 1,945 website visitors on launch day is approximately **$89,350.97**.

Would you like further details on any part of the calculation, or have any specific questions? 

Here are 5 related questions for deeper exploration:

1. How would changes in the probabilities affect the expected sales value?
2. What if the number of visitors increases by 10%—how would that change the expected total sales?
3. What is the contribution of each shoe type to the overall expected sales value?
4. How would introducing a new type of shoe affect the expected value calculation?
5. How can expected value analysis be useful in other areas of business decision-making?

**Tip:** Understanding how probabilities work in expected value calculations helps you make more informed predictions when dealing with uncertain outcomes.

A shoe company conducts a survey to determine the expected value of online sales for their new line of shoes coming out next fall. Based on past years, they have observed the following data on the probability of selling different types of shoes in the new line. The company expects that 1945 people will visit the website for their new line on launch day. Note that some of the online shoppers will not make a purchase. Sneakers $53.99 3/25, high heels $71.25 4/15, sandals $39.50 6/25, loafers $45.75 6/25.

Calculate the expected value of online sales for a shoe company's new line launch, using probability and past sales data. The problem covers key statistical concepts like expected value and probability, with detailed calculations of potential earnings from 1,945 website visitors.

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