# Counterfactuals And Woodchuck Probability Distributions

I have, with mechanical aid, again winged my way across the globe in search of treasure. I landed disoriented and would not trust myself to calculate even the integral of the exponential function. I also cannot come up with an original post.

But since we left off with the Best Statistical Question Ever, I thought it best to confuse matters further by discussing probability distributions over counterfactual events.

Example: if a woodchuck could chuck wood, how much wood could a woodchuck chuck?

This is an unknown quantity and therefore must be represented by a probability distribution. Which distribution is the correct one? Is there only one correct one?

The first clue something is amiss is the chucking units are not given. But we can assume them: for the sake of this exercise use face cords.

A face cord, according to the site Northline Express is,

The two most common ways to measure firewood is by the “Full Cord” or “Face Cord”. You will see that our racks refer to a “Face Cord”. A face cord measures 8′ wide x 4′ high, the depth of a face cord may vary depending on the person who is cutting the wood (average firewood length is 16″-18″). A “Full Cord” of firewood measures 8′ wide x 4′ high x 4′ deep making it difficult to determine what size firewood rack to purchase since firewood is cut to different depths. Due to the confusion of calculating full cords we have decided to calculate our firewood racks by the face cord. Please check to ensure that you are purchasing the proper size firewood rack.

Your own author was responsible for creating from the raw, by hand with a fifteen-pound sledge and steel wedge, several score of face chords when he was a teen under the direction of his father. Incidentally, anybody who in the comments re-tells the old joke about the number of times wood heats you will earn permanent banishment.

But knowing the unit tells use nothing about woodchucks. Remembering that all probabilities are conditional on explicitly stated evidence, and that different evidence can lead to different probabilities, your homework is to describe the probability distribution representing the wood chucking ability of woodchucks assuming the counterfactual that woodchucks can chuck wood.

Be clear in your definitions! Best of luck. And see you when I wake up.

## 20 Thoughts

1. Speed says:

To give some perspective to our host’s labors with a fifteen-pound sledge … The weightiest offering from Lowes is a ten pounder. Home Depot (obviously a manlier retailer) has several 16 pounders.

I leave it to the reader to determine which requires the least work input from the operator to split a cord (or any other arbitrary amount) of wood. Is there an advantage to swinging a big stick?

Since the job of cord wood is heating your rustic log cabin, heat content of the split timber is key. This chart may be useful to the most careful shopper. The range is considerable.
http://www.consumerenergycenter.org/home/heating_cooling/firewood.html

Me? I’m going to write to my congressman supporting any and all legislation that surfaces more and cheaper natural gas.

2. RC Saumarez says:

A woodchuck’s chucking ability is measured in microcords (~0.2 cu inches).inches/hour. The probability distribution of the instantaneous chuck is a function of volume of standard density wood in microchords(m), distance chucked (d) and duration of continuous chucking time (T). Preliminary research suggests that it may be approximated by:

Gamma*(a,b,T,x)=(e^-bT)^-a x^(a-1) e^(-e^bT x))/Gamma(a)

This is a modified gamma function that has a peak that draws closer to zero as T increases, representing an increasingly fatigued woodchuck ( if you use “Mathematica”, set b=exp(-t) and evaluate [GammaDistribution[a,b],x]. The probability distribution is of the form:

p(m,d,T)=Gamma*(a0,b0,T,m).Gamma*(a1,b1,T,d)/Q(a0,b0,T,a1,b1)

where Q normalises the distribution.

[Interested readers could apply for funding to evaluate how this distribution is likely to change]

3. RC Saumarez says:

Sorry, I meant … how this distribution is likely to change with Global Warming.

4. Gary says:

Mr. Briggs: please be aware that it is illegal to impersonate a corder of wood in the State of Rhode Island (http://www.rilin.state.ri.us/Statutes/TITLE11/11-14/11-14-2.HTM) and doing so will make you liable to a fine of \$20-100. Otherwise, we have woodchucks in abundance — perhaps because of 5 cents per nose bounty no longer is paid — so please feel free to visit to observe the chucking. Or, rather, to test empirically the estimates of the probability of chucking.

5. Ken says:

Whatever equations, distributions, etc. are developed are conditional on the definitions of terms used, and, associated assumptions:

– First & most basic: IF a woodchuck Could chuck would, would the woodchuck Choose to chuck it? Woodchuck psychology & stamina could be a factors, or ignored — either way this must be stipulated. If the model developed assumes that a wood-chucking-capable woodchuck would not choose to chuck (i.e. would choose to not chuck) wood then the answer is exactly zero. At least for that critter. Population variances c/would factor in.

– Alternatively, if one assumes the woodchuck would choose to chuck then a host of other assumptions come into play, such as for how long, are breaks allowed (e.g. one might apply criteria stipulated by comparable law, or, comparable union-negotiated rates), etc. etc. Presumable a Republican woodchuck (if we’re making assumptions party affiliations is fair game….which leads to another assumable factor, below) would out-chuck, based on initiative and intrinsic values, the wood chucked by a Democratic woodchuck coddled under union-negotiated rate scales.

RE “fair game” — this naturally begs the question: In this scenario will woodchucks be “fair game” subject to hunters and/or other predators? If yes, in-vs.-out of season variations arising from sudden death/disappearance associated with calendar dates would complicate the modeling — presumably flattening the distribution when random death is more likely.

The following might serve as a mental stimulant for such considerations, which clearly are not confined to just woodchucks: http://www.youtube.com/watch?v=F6etfJgZQ7A

6. JH says:

I can’t do the homework without data. No data, no probability/statistical modeling.

Whatâ€™s the old joke about the number is of times wood heats? What is the number? One is a lonely number. Definitely not 3. Three has never liked me since we met in first grade. Four rhymes with death in Chinese. All rankings (with 1 being the best) higher than 5 are not acceptable. So, TWO is my answer.

7. DAV says:

There’s a woodchuck living under my garage. It had studiously avoided chucking ANY wood as far as I can tell so the answer is likely 0 which is pretty close to MY average. Isn’t this post a bit premature for Feb 2?

8. Doug M says:

A woodchuck would chuck as much wood as a woodchuck could chuck if a woodchuck could chuck wood.

RC Saumarez is off to a good start modeling efficiency of woodchuck chucking over the time spent chucking.

The missing variable is how much time would be spent chucking vs. non-chucking activity — eating, sleeping… climatology.

Climatology? Don’t forget that the woodchuck is also known as the groundhog, and is the one of the most famous weather prognosticators in the land.

Methinks a certain statistician-meteorologist is jealous of the woodchucks fame.

9. The heart of the exercise is to describe the probability distribution representing the wood chucking ability of woodchucks assuming the counterfactual that woodchucks can chuck wood.

I think that perhaps a few of our friends above are overthinking the task. Like most human and woodchuckian abilities, the probability distribution of woodchucking abilities will be roughly a normal (Gaussian) distribution about a mean which would have to be determined by observation. You can’t just make this stuff up.

However, I will do a bit of speculating. If the reward to the woodchuck (whether food, gold or near worthless Greenbacks) is based on performance, there will be EEOC complaints filed by the ACLU, SPLC, ACORN and other such parasites who believe that the lower-performing woodchucks are victims of past discrimination based on race, creed, religion, color, national origin, sex, gender, sexual orientation, LGBT status, addiction, handicap, bad mothers, poor public schools, and “other.” The differences in performance can in no way be attributed to differences in innate ability or work ethic.

Rewards can never be based on a normal (Gaussian) distribution just because some woodchucks can chuck more wood. Those on the left side of the curve will need to be boosted through government programs. Those on the right will count themselves lucky if the only penalty they pay for their ability and hard work is “progressive” taxation.

Eventually, the normal (Gaussian) distribution will shift to the left as the high producers lose motivation and spend more time in Cabo with their families and less time chucking.

QED

10. John M says:

Once you’ve figured out how much wood a woodchuck can chuck, you can start working on how much ground a groundhog could grind if a groundhog could grind ground.

Which reminds me of why my coffee tastes like mud. (It was ground this morning.)

That ought to get me banned without talking about heating wood.

11. Joe says:

I assume it would be log-normal….

12. Alan Bates says:

How many times does wood heat you?

O.K. I’ll take the Bates bait …

3 times.

1) When you cut it down and/or split it with a 6.8 kg hammer.
2) When you burn it
3) When the CO2 contributes to global warming before being sequestered again into new wood. But this must be countered by the opposite effect of the smoke and/or clouds created.

Of the three I suspect 1) and 2) are positive. 3) who knows – can I have a research grant to find out …

Joe “log-normal” is dreadful – but you know that …

13. Alan Bates says:

Bates. Appears my use of strikethrough in HTML doesn’t work …

14. Briggs says:

Joe,

A+!

15. Will says:

This is an easy one.

Determine the amount of force a woodchuck and apply each day. Multiply that by the maximum number of days a woodchuck can live. That will give you the upper bound amount, in weight, of wood that the woodchuck can displace.

Nobody said the woodchuck Couldn’t be fed through an IV, and that the wood couldnt be in dust form. 🙂

16. RC Saumarez says:

@Briggs @Joe

Distribution of what?

17. Rich says:

I’d say it’s the same answer as the answer to the question, “Where does an 800lb gorilla sleep?”

18. Clayton W says:

But isn’t chucking wood a bbarbon increase? Where would the woodchuck get the carbon offset? Since we know that all the global warming is anthromorphic and nature abhors man, we can safely assume that that a woodchuck will chuck NO wood.

Can we chuck Gore?

19. David S says:

Doug M I am not sure if I follow you. Why would a woodchuck, if it could chuck wood, chuck as much wood as it could? Are you assuming predestination of woodchuck behaviour? If so does that apply to all woodchucks or just the Prof’s notional one?