Expected Number Of Kids And Cost Calculations For Elsa's Par

Expected number of kids and cost calculations for Elsa's party

Elsa's birthday party at the Young Chefs Academy (YCA) involves several cost factors based on the number of children attending. Dorothee must decide on the number of kids to quote to YCA one week in advance, which determines the initial cost. The cost structure is as follows:

• The YCA charges $18 times the quoted number of kids, regardless of actual attendance. If fewer children show up, no refund is given.
• If more children attend than the quoted number, Dorothee pays an additional $25 per extra child.
• Goodie bags are purchased one week prior at $5 each. After the party, any leftover bags can be sold back at $3 each to Mary. If more bags are needed on the day, Dorothee buys extra at $8 per bag.

Elsa invites 20 friends, and data on the number of children who might attend, along with their probabilities, is provided.

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To analyze the expected attendance and costs associated with Elsa's birthday party, we first consider the probabilistic distribution of the number of children attending, based on the data provided. The goal is to calculate the expected number of children, total costs under different scenarios, optimal ordering amounts, and risk assessments such as the probability of running out of goodie bags.

A. Expected Number of Kids Attending

Given the probability distribution of children attending out of Elsa's 20 invited friends, the expected number of kids is calculated as the sum of each possible attendance multiplied by its probability. This measure provides a statistical average of children expected to show up.

Assuming the distribution data indicates various attendance probabilities (e.g., 14, 15, 16, 17, 18, 19, 20 kids with respective probabilities), the calculation follows:

Expected number (E) = Σ (number of kids) × P(number of kids)

Depending on the distribution, this may result, for example, in an expected attendance of approximately 16.5 children.

B. Cost When 1 Extra Kid Attends

Suppose Dorothee reports 17 children and purchases 17 goodie bags, but in reality, 18 children attend. The total cost includes:

• YCA cost: 18 children × $18 = $324
• Additional children beyond quoted number: (18 - 17) = 1, costing $25
• Goodie bags: 17 purchased, but 18 children attended – so, need to buy 1 extra bag at $8, unless the extra bag is bought in advance or needs to be bought separately. Since bags are already purchased, the cost remains as is. But in this scenario, the initial purchase is 17, so she needs to buy one at $8 to accommodate the extra child.

Total cost = $324 (YCA) + $25 (extra kid penalty) + $8 (extra goodie bag) = $357

C. Cost When Fewer Kids Show Up

If 17 goodie bags were purchased assuming 17 children but only 14 arrive, the costs involve:

• YCA cost: 17 × $18 = $306
• Goodie bags: 14 children attended; 17 bags purchased, so 3 leftover
• Mary buys back 3 goodie bags at $3 each: 3 × $3 = $9
• Remaining 17 - 14 = 3 unneeded bags; Dorothee keeps these, so no extra purchase needed.

Thus, total cost: $306 + costs for unused bags (netting) could be viewed as the initial purchase minus Mary’s buy-back, leading to an effective cost of $306 plus the potential for leftover management costs. Her net expenditure is effectively $306 minus income from Mary, resulting in: $306 - $9 = $297.

D. Expected Total Cost and Goodie Bags Sold Back

Assuming Dorothee reports 17 children, the expected total cost involves integrating over all possible actual attendance figures, weighted by their probabilities. The cost calculation accounts for:

• YCA charges based on actual attendance, which varies probabilistically.
• Goodie bags purchased: 17 in advance.
• Expected number of goodie bags sold back to Mary: sum of (number of bags left after actual attendance) times their probability, with the assumption that goodie bags are bought at 5 dollars and sold back at 3 dollars.

Probability of stock-out (not enough goodie bags) can be derived from the cumulative probability that actual attendance exceeds the purchased amount. The fill rate quantifies the proportion of goodie bags supplied from the initial purchase.

The probability that total costs exceed $400 can be calculated by considering the distribution of costs over all possible attendance scenarios and summing the probabilities where costs surpass this threshold.

E. Optimal Quoted Number to Minimize Expected Cost

If Dorothee sets the quote equal to the number of goodie bags she plans to buy, she minimizes the expected total cost by choosing the number that balances the cost of over-ordering against the penalty of under-ordering. This involves solving the optimization problem:

minimize over n the expected total cost function, which considers the fixed costs, penalty costs for over- and under-attendance, and the purchase cost of goodie bags. The optimal n can be found by analyzing the cost function relative to different values and choosing the one with the lowest expected cost.

F. Number of Goodie Bags to Prevent Running Out with 90% Confidence

To ensure less than a 10% chance of running out of goodie bags, Dorothee needs to purchase a number of goodie bags exceeding the (90th percentile) quantile of the distribution of attendance. This value can be derived from the cumulative probability distribution, choosing the smallest number of bags for which the probability of needing more exceeds 10%.

G. Spending If She Could See the Future

When Dorothee knows exactly how many kids will attend, her total cost calculation simplifies to:

• YCA cost: actual attendance × $18
• Goodie bags: purchase exactly the number needed (without surplus or shortage costs)
• Cost of any additional purchases if needed (which in this perfect knowledge case, is zero)

The expected total expenditure is the sum over all possible attendance numbers of the exact cost times the probability of that attendance, effectively the sum of costs for each attendance scenario weighted by its probability, which reduces essentially to the cost of the actual attendance.

H. Expected Mismatch Cost

The mismatch cost when reporting 17 kids but actual attendance varies is calculated as the expected penalty associated with the difference between the quoted number and actual attendance. This includes costs for excess or shortage of goodie bags and overpayment penalties for extra children, weighted by the probability of each attendance number, and considering the purchase and resale prices of goodie bags.

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