A store owner stocks an out-of-town newspaper that is...

A store owner stocks an out-of-town newspaper that is sometimes requested by a small number of customers. Each copy of this newspaper costs her 70 cents, and she sells them for 90 cents each. Any copies left over at the end of the day have no value and are destroyed. Any requests for copies that cannot be met because stocks have been exhausted are considered by the store owner as a loss of 5 cents in goodwill. The probability distribution of the number of requests for the newspaper in a day is shown in the accompanying table. If the store owner defines total daily profit as total revenue from newspaper sales, less total cost of newspapers ordered, less goodwill loss from unsatisfied demand, what is the expected profit if four newspapers are order? $$ \begin{array}{lcccccc} \hline \text { Number of requests } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & 0.12 & 0.16 & 0.18 & 0.32 & 0.14 & 0.08 \\ \hline \end{array} $$

A store owner stocks an out-of-town newspaper that is sometimes requested by a small number of customers. Each copy of this newspaper costs her 70 cents, and she sells them for 90 cents each. Any copies left over at the end of the day have no value and are destroyed. Any requests for copies that cannot be met because stocks have been exhausted are considered by the store owner as a loss of 5 cents in goodwill. The probability distribution of the number of requests for the newspaper in a day is shown in the accompanying table. If the store owner defines total daily profit as total revenue from newspaper sales, less total cost of newspapers ordered, less goodwill loss from unsatisfied demand, what is the expected profit if four newspapers are order?
$$
\begin{array}{lcccccc}
\hline \text { Number of requests } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.12 & 0.16 & 0.18 & 0.32 & 0.14 & 0.08 \\
\hline
\end{array}
$$

Key Concepts

Expected Value

The expected value is a fundamental concept in probability that involves calculating a weighted average of all possible outcomes, where each outcome is weighted by its probability. In the context of business decisions such as inventory management, it is used to estimate the average profit, cost, or loss over a range of different demand scenarios.

Probability Distribution

A probability distribution describes the likelihood of various possible outcomes. Here, it determines the chance of receiving a certain number of newspaper requests. This distribution is crucial for calculating the expected value of demand and consequently for making sound inventory decisions.

Newsvendor Model

The newsvendor problem is a classic model in operations management and inventory theory that addresses how many units of a product to order when facing uncertain demand and a limited selling period. The model balances the costs associated with unsold items (overage) and missed sales opportunities (underage), and is directly relevant to situations where leftover stock has no salvage value.

Cost-Benefit Analysis

Cost-benefit analysis in this context involves comparing the costs of ordering newspapers (purchase cost) against the benefits of selling them (sales revenue) and the additional costs associated with unsatisfied demand (goodwill loss). This analysis helps in determining the optimal order quantity to maximize expected profit.

Goodwill Loss

Goodwill loss refers to the negative impact, quantified monetarily, that unsatisfied customer demand can have on a business's reputation and future sales. In this problem, each unmet newspaper request results in a stated loss, which must be factored into the overall profit calculation, highlighting the importance of considering both direct and indirect costs in decision-making.

Transcript

00:02 In the current problem, the first thing they ask that what does x represent? now i will tell you why i have all the answers written just next to it because it's a big question, eight subparts to it, and if i start writing each and everything, then it will take a lot of time.

00:22 So i'm trying to keep it as quick as possible.

00:27 So the first thing is x, what is x? so in the question, if you see that they have given the probability distribution of the number of videos rented by the customers.

00:43 That's why x would mean the number of dvds rented by any customers.

00:49 That means what a particular customer can rent one, zero dvds, that means no dvd at all.

00:57 And that also has a probability attached to it.

00:59 You know they come and they don't find it or they don't like it whatever and then other option being like renting one then two and three then four then five so that's why x is the number of dvds okay not number of people coming to the store or something it's the number of dvds next they're asking what is the probability of x being three so if you see the total probability of over here should always be 1.

01:31 But if we add this up, okay, we will see it is coming to 0 .88.

01:37 That means what the remaining 1 minus 0 .88 should be the probability for x equals to 3.

01:50 Next they are asking what is the probability that they rent at least 4 dvds.

01:57 At least 4 means either they will rent one, four dvds or they will rent five dvds.

02:07 So what are the probabilities attached to it? one is this, the other one is this.

02:11 So this is 0 .07 plus 0 .04.

02:15 That's the answer.

02:16 Next is what is the probability that at most three dvds, at most three dvds.

02:23 Think of the expression.

02:24 Maximum they will rent three.

02:27 So, they can rent three that is a maximum or less than it, which is two or one or zero.

02:35 Now, if you see, okay, if i clear the screen a little bit for you, they want to know the probability of this, which would mean total probability minus this amount.

02:49 Correct? they want the total probability of this, which is this, summation of all this four.

02:55 Instead, we can think since total probability is already one minus this.

03:01 Now, why is a different approach? you know, why am i taking this approach? because most importantly, we need to understand the importance of the probability of a plus probability a complement is equals to 1.

03:21 Correct so if this particular event is a then this will be a complement okay and in probability again and again we will come across this feature so this formula over here is really important and that's a reason they have given this example okay so that we get familiar with it next they're asking that what would be the expectation of the first variable videos to go and then the same question they will ask for the second one as well, entertainment headquarters.

04:05 So we know to find probabilities, it will be value into probability, correct? so we have calculated that value into 1 into 0 .5 is 0 .5, 2 into 0 .24 is 0 .48, 3 into 0 .1 is 0 .36, 0 .7 into 4, 4 into 0 .07 is 0 .28 and 5 into 0 .4 is 0 .2...

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