A factory makes bags of six assorted filled chocolates. Each bag contains two each of three varieties; they always have caramel-filled chocolates, have either chocolate cream or vanilla cream, and have one of orange, raspberry, or mocha. The assembly line is set up so half the bags start at the chocolate cream machine and half at the vanilla cream machine, receiving two chocolates; they merge to a single caramel machine, getting shuffled in the process; and finally they split randomly into three lanes that lead to machines dispensing orange, raspberry, and mocha-filled chocolates respectively.
There are many questions one can ask about counting: how many ways are there to fill a bag of chocolates (i.e., how many distinct lists of varieties might a bag contain)? How many ways to fill a bag include orange-filled chocolates? How many ways include both chocolate cream and mocha-filled chocolates? How many ways include neither vanilla cream nor raspberry-filled chocolates?
The question I’m more interested in discussing, however, is this: Suppose something is going wrong in the assembly line and over half the bags are coming out overfilled. Before you break open the chocolates and look at their insides, form a hypothesis as to what has gone wrong. What machine or machines are most likely to be malfunctioning? Why?
Your answer consists of your assumptions and the arguments based on them, with enough context restated from the problem that the answer reads smoothly.
There are a few conflicting but appropriate assumptions you could make here. The first is that either the malfunction is consistent, affecting all bags that come through that machine, or it’s inconsistent, affecting at most all of the bags, but possibly fewer.
The second choice of assumptions is whether the description “over half” implies “not all.” If every bag were coming out overfilled, you’d expect that to be specified… unless you’re a logician teaching a math class, in which case “over half” definitely includes the possibility of “all.”
But the important thing for writing up an answer is to be clear on what you’re assuming above and beyond the information given in the problem.
There is a third assumption that I expected everyone to make: that the most likely explanation involves malfunction of the fewest machines. It may not always be the case, but we’re assuming the candy factory is run well enough that machines are more likely to be working properly than improperly.
So what’s your answer?