I stumbled into a strange dilemma yesterday while I was throwing an old receipt into the trash. And since I have managed to complete my work early today, I thought that we could discuss it at length here.

And by that, I of course mean that I could ramble insanely for thirteen hundred words, and you can read it all because you have nothing better to do.

The central issue at the heart of the dilemma is the garbage can. It was fairly messy and garbageful yesterday, and for future reference, I shall assume that it was a standard Yucky Garbage Can. It had all the standard criteria covered…

• Wetness – I had washed the dishes earlier and anything caught in the drain was deemed to be garbage.
• Oiliness – Congealed fat and oil from a leftover dish that I didn’t want to clog my pipes with.
• Recently slain organism – A moth I had crushed earlier with a tissue.
• Rotten flesh – One of my peaches tasted funny and had a weird looking spot on it.
• Smell – Standard two or three day old garbage smell.
• Used syringes – A given, found in any standard garbage can.
• Unknown – I had relatives over recently and they may have deposited articles of unknown nature and origin into the garbage.

So for all intents and purposes, we can assume that the standard YGC fits this criteria, and that the garbage can of interest I was dealing can be considered to be a YGC. The purpose of identifying a YGC and identifying mine as such is to establish the garbage can as something you would not regularly want to put your hand into.

Now here’s how it all started.

Whenever I exchange money for goods and services, there exists a significant possibility of change being returned. Particularly when the dollar value of the goods or services is less than the denomination of the least valued dollar bill accessible to me at the time of exchange. Now this “change” is – through years of practice and habit – generally stored in my right front pocket. (Unless I am performing a purchase without the use of pockets or pants, in which case I will just grasp the coins in my fist until I can find somewhere to deposit them.) However, the right front pocket is not exclusively used for the storage of change, and there are frequent occasions when it will be asked to store more than just coins.

This was one such instance, when there was a varied assortment of coins in my pocket along with an old receipt.

When I came home last night, in an attempt to throw out the receipt, I had also inadvertently grabbed a single coin in the same motion. It’s possible that the receipt was folded or crumpled, and that a single coin had become trapped within that fold or crumple…but that’s something I’ll never know for sure. As fate would have it, I didn’t notice the extra coin in the intended throw-out-pay-off until I actually threw the handful of receipt into the garbage and heard the unmistakable sound of coin against garbage can.

I hadn’t noticed…until it was too late to notice.

Judging from the sound of coin/garbage can impact, and taking into consideration the differences found in the comparisons of the present count of coins remaining in the pocket and the probable count of coins that were estimated to originally be in the pocket, I narrowed down the possibilities and concluded that the coin could have been anything between a quarter to a twoonie. So possibly a loonie as well.

And therein lies the dilemma. I am presented with an undesirable challenge (rooting barehanded through a YGC) for a desirable – and variable – reward (a quarter, loonie, or twoonie). And it’s been something I’ve been thinking about ever since.

Now any risk management course will teach you to weigh the possible costs against the possible gains to find an acceptable value of risk that will tell you whether or not to go for it. And for that, we require a calculus of values. To determine the cost (by determining how yucky the garbage really is, and how much yuck-tolerance your hands have) and determining the gain (the number of coins, the possible value of each coin, and the probabilities of those respective values for each coin).

In my case, since I was only able to narrow it down to three possible coins all with equal possibilities, we can calculate the expected value of the single, unknown coin. Recalling OAC Finite with Mr. Esche, the expected value can be determined as follows…

…where E(X) is the expected value, and p_i and x_i represent the probability and value of each possible outcome, respectively.

Thus, we find…
E(X) = (1/3)$0.25 + (1/3)$1.00 + (1/3)$2.00 = $1.08333…
…which we round to $1.00, since – despite my angry letters – the Canadian mint has yet to issue a one point zero eight three repeat dollar coin.

We can also get more specific in dealing with general cases. Perhaps we establish some rules – the cost of rooting around for anything more than three coins is too steep, and a reward of anything under fifty cents is too little. We can then outline our risk behaviour with this graph…

…in what I call the Coin in Garbage Rescue Ratio or CiGaR Ratio for short. Each type of coin had an entered value, with quarter-sized increments also added between $0.25 and $2.00. The rest can be ignored, because when you get north of a dollar, we assume that dimes and below are essentially meaningless. If a combination of coin value vs. coin quantity is found in the region between the function and the x-axis, then rescue is advisable.

The first things to note are the maximums found at x = $0.75 and $1.50, where y = 3. This is because those are the only values that can be made with exactly three acceptable coins. In comparison, we see that x = $1.75 yields a zero value. Not because $1.75 isn’t a sufficient reward, but because there’s no way to build it from three or less coins, and thus the cost is too high. A coin has to be sacrificed to lower the cost back to the maximum allowed of three (ie. leave a quarter, so x = $1.75 – $0.25 = $1.50, which has a corresponding non-zero y value).

I’m not really sure why there’s a negative region at 17 cents, though. I’m not really sure what a negative CiGaR Ratio represents. Negative rescue? So…the opposite? Is the graph telling us that in the event we have 17 cents on us, it’s actually beneficial to throw it away? It’s an unexpected finding, but I suppose things like that should be expected whenever we are exploring new ideas, and we must remain open minded in our pursuit of knowledge.

Maybe at some point we can figure out how to factor in inaccurate assessments of coins – for instance, depending on the actual material of the garbage can, the inflection and tone of coin bouncing off it at various trajectories and velocities, the actual size and thus monetary value of the coin maybe be suspect to doubt and different probability models. This may have come in particularly handy in my case, when I discovered – after plunging my hand into the YGC in a CiGaR Ratio advised rescue attempt – that what I had actually dropped was a nickel. Which actually broke the case studies and seems to defy CiGaR Ratio conventions. However, I still believe the system works – it’s just that in my specific case, the parameters of the case just weren’t correct, which isn’t a fault of the system but a fault of the user.

It’s something to think about if you ever find yourself in a similiar situation.

“No. It’s been two weeks and I don’t feel any different. All I’ve done is gain eight pounds. Now, what’s in this?”
“Oh, a little of this and a little of that. Open.”
“It’s sugar isn’t it? I’M IN THE PLACEBO GROUP!”