Extracting information correctly from text I am stuck with interpreting the information given in the following exercise:
Due to a recent increase in the price of tomatoes a pub landlord is looking to reduce the quantity of tomatoes he buys, but wants to minimize the risk of running out of it on any one day. Data collected over the last 20 days has shown that on average 8 customers order tomatoes and that the average number of orders per day is 25. The standard deviation of the data collected was 1.2. Calculate the probability that the landlord will run out of tomatoes if he buys enough for 10 portions per day.
My first interpretation: I would model this using the random variable $X = $ number of tomatoes ordered. $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. Then from the data I have an estimate $\bar X = 8$ for $\mu$ and $S = 1.2$ for $\sigma$, and I know that
$$
\frac{\bar X - \mu}{S/\sqrt{20}} \backsim t_{19}
$$ 
But I am not sure this helps me to find the required probability $P(X \ge 10)$ and also I haven't used the information that $25$ meals are ordered per day on average.
My second interpretation: Let $\pi$ be the proportion of meals with tomatoes ordered per day. Then the data gives us an estimate $p = 8/25 = 0.32$ and we know that 
$$
\frac{p - \pi}{\sqrt{\frac{\pi(1 - \pi)}{20}}} \backsim N(0,1)
$$
but again I don't think this is what I should look for as it would not be useful in calculating the required probability.
My third interpretation: Let $Z$ be the number of meals with tomatoes ordered per day. Then $Z \backsim \text{Bin}(20, 0.32)$ from which we can compute the probability $P(Z \ge 10)$ as required. However, here I don't think the random variable accurately reflects the given information, in particular treating the number of observations as repeated attempts sounds weird, and again I haven't used given information (namely the sample standard deviation).
I am quite confused and not sure how to progress -- all these interpretations seem to be not entirely correct. Any hint would be very useful, thanks a lot for your help !!
 A: You're right to be confused. The exercise is unclear on two points:


*

*How is the mean daily count of customers who order tomatoes (8) supposed to be reconciled with the mean daily count of orders for tomatoes (25)? Is it that a given customer can make more than one order? If so, isn't the number of customers irrelevant, and only the number of orders matters? For that matter, is it guaranteed that only one tomato is needed per order? And what quantity is 1.2 the SD of?

*What exactly are we trying to estimate? Is a portion the same thing as an order? Do we want the probability that the landlord will run out of tomatoes within a single day, or over a longer period? Do unused tomatoes carry over?
You could try asking your instructor.
Maybe the sanest way to interpret the question is as follows. Over 20 days, the daily occurrence of an instantaneous event was counted. The daily counts have a mean of 25 and an SD of 1.2. What is the probability that the count of events on a given day will exceed 10? (I have completely ignored the 8.)
A natural way to approach that problem is to construe the daily counts as i.i.d. draws from a negative binomial distribution. Estimate the parameters and then use the cumulative distribution function to compute the probability that a new draw will exceed 10.
