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The image you uploaded presents an example related to inventory management and probability analysis for a bakery that bakes 100 cakes daily. The demand for cakes follows a probability distribution, and certain costs and penalties are given for both production and unmet demand.

### Key information extracted from the image:

1. **Probability Distribution for Cake Sales:**
   - Sale of 97 cakes: Probability = 0.10
   - Sale of 98 cakes: Probability = 0.15
   - Sale of 99 cakes: Probability = 0.20
   - Sale of 100 cakes: Probability = 0.35
   - Sale of 102 cakes: Probability = 0.15
   - Sale of 103 cakes: Probability = 0.05

2. **Cost and Pricing Details:**
   - **Variable production cost per cake:** ₹14
   - **Selling price per cake:** ₹18
   - **Penalty for each unsold cake:** ₹3
   - **Penalty for each unit of unmet demand:** ₹1

### Objective:
The bakery needs to decide how many cakes to bake in order to balance the costs, selling prices, and penalties based on the demand probabilities. This type of problem involves analyzing expected costs and profits based on different production levels, demand scenarios, and associated penalties.

#### Potential Steps to Solve:
1. **Expected Profit Calculation** for each scenario of cake production and demand using the following elements:
   - Total revenue from cake sales.
   - Cost of production for the cakes.
   - Penalty for unsold cakes (if production exceeds demand).
   - Penalty for unmet demand (if demand exceeds production).

2. **Optimization**: Find the optimal number of cakes to produce to maximize profit or minimize cost.

Would you like a detailed step-by-step solution to this problem?

Here are five related questions you might explore:
1. How do we calculate expected profit for different levels of cake production?
2. What happens if the bakery produces more cakes than the demand?
3. How do penalties affect the bakery’s overall profit?
4. How can we use this probability distribution to make decisions under uncertainty?
5. How do changes in costs or prices impact the optimal production strategy?

**Tip:** In problems with penalties for overproduction and unmet demand, balancing production based on the expected demand is key to maximizing profits.

A bakery bakes 100 cakes per day. The sale of cakes depends on demand with the following probability distribution. Given the cost of production, selling price, and penalties for unsold and unmet demand, determine the optimal number of cakes to bake to maximize profit.

Analyze the probability distribution of cake sales in a bakery and calculate the expected profit. The solution involves cost optimization based on production costs, selling price, and penalties for unsold cakes or unmet demand. Learn how to apply probability and expected value to solve this real-world problem.

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