8.1.3: Berger's Burgers

Last updated
Save as PDF

Page ID: 51669

Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

A former 6.050J/2.110J student opened a fast-food restaurant, and named it in honor of the very excellent Undergraduate Assistant of the course. At Berger’s Burgers, meals are prepared with state-of-the-art high-tech equipment using reversible computation for control. To reduce the creation of entropy there are no warming tables, but instead the entropy created by discarding information is used to keep the food warm. Because the rate at which information is discarded in a computation is unpredictable, the food does not always stay warm. There is a certain probability, different for the different menu items, of a meal being “COD” (cold on delivery).

The three original menu items are Value Meals 1, 2, and 3. Value Meal 1 (burger) costs $1, contains 1000 Calories, and has a probability 0.5 of arriving cold. Value Meal 2 (chicken) costs $2, has 600 Calories, and a probability 0.2 of arriving cold. Value Meal 3 (fish) costs $3, has 400 Calories, and has a 0.1 probability of being cold.

Table 8.2: Berger’s Burgers
Item	Entree	Cost	Calories	Probability of arriving hot	Probability of arriving cold
Value Meal 1	Burger	$1.00	1000	0.5	0.5
Value Meal 2	Chicken	$2.00	600	0.8	0.2
Value Meal 3	Fish	$3.00	400	0.9	0.1

There are several inference questions that can be asked about Berger’s Burgers. All require an initial assumption about the buying habits of the public, i.e., about the probability of each of the three meals being ordered $p(B)$, $p(C)$, and $p(F)$. Then, upon learning another fact, such as a particular customer’s meal arriving cold, these probabilities can be refined to lead to a better estimate of the meal that was ordered.

Suppose you arrive at Berger’s Burgers with your friends and place your orders. Assume that money is in plentiful supply so you and your friends are equally likely to order any of the three meals. Also assume that you do not happen to hear what your friends order or see how much they pay. Also assume that you do not know your friends’ taste preferences and that the meals come in identical packages so you cannot tell what anyone else received by looking.

Before the meals are delivered, you have no knowledge of what your friends ordered and might assume equal probability of 1/3 for $p(B)$, $p(C)$, and $p(F)$. You can estimate the average amount paid per meal ($2.00), the average Calorie count (667 Calories), and the probability that any given order would be COD (0.267).

Now suppose your friend Alice remarks that her meal is cold. Knowing this, what is the probability she ordered a burger? (0.625) Chicken? (0.25) Fish? (0.125). And what is the expected value of the amount she paid for her meal? ($1.50) And what is her expected Calorie count? (825 Calories)

Next suppose your friend Bob says he feels sorry for her and offers her some of his meal, which is hot. Straightforward application of the formulas above can determine the refined probabilities of what he ordered, along with the expected calorie count and cost.