The following question comes from [*Mathematical Statistics with Applications (7th ed.)*](https://rads.stackoverflow.com/amzn/click/com/0495110817) by Wackerly, Mendenhall, and Schaeffer:

> 
> A merchant stocks a certain perishable item. She knows that on any given day she will have a
> demand for either two, three, or four of these items with probabilities $.1$, $.4$, and $.5$, respectively.
> She buys the items for $\$1.00$ each and sells them for $\$1.20$ each. If any are left at the end of
> the day, they represent a total loss. How many items should the merchant stock in order to
> maximize her expected daily profit?
> 
> 
> 

The answer provided in the book says that **three** items should be bought, but I don't understand the reasoning in the manual, highlighted in **bold**:

> 
> There are three scenarios:
> 
> 
> * if she stocks two items, both will sell with probability $1$. So, her profit is $\$0.40$.
> * if she stocks three items, two will sell with probability $.1$ (a loss of $.60$) **and three
> will sell with probability $.9$**. Thus, her expected profit is $$(–.60).1 + .60(.9) = \$0.48$$
> * if she stocks four items, two will sell with probability .1 (a loss of $1.60$), three will
> sell with probability $.4$ (a loss of $.40$), and four will sell with probability $.5$ (a gain
> of $.80$. Thus, her expected profit is $$(–1.60).1 + (–.40).4 + (.80).5 = \$0.08$$
> So, to maximize her expected profit, stock three items.
> 
> 
> 

This value of $0.9$ is simply the probability of buying three items ($0.4$) added to the probability of buying four items ($0.5$)... I don't understand how this is justified. My thinking is the probabilities of more items being sold than bought should be discounted, leaving an $\frac{0.4}{0.4+0.1}=80\%$ chance of selling all three of the bought items and a $\frac{0.1}{0.4+0.1}=20\%$ chance of selling only two of them. This would make the expected profit from buying three items equal to $$ (-0.6)(0.2)+(0.6)(0.8)=\$0.36 $$

whereby buying **two** items maximizes the daily profit. Wouldn't the answer provided in the manual be more appropriate for the question

> 
> A merchant stocks a certain perishable item. She knows that on any given day, **she will sell at least three items with probability $0.9$** and four items with probability $0.5$.
> She buys the items for $\$1.00$ each and sells them for $\$1.20$ each. If any are left at the end of
> the day, they represent a total loss. How many items should the merchant stock in order to
> maximize her expected daily profit?
> 
> 
>