comes in drunk 12 times per annum
= once a month
weekly testing = 4 ish times per month
20 ish working days per month
odds of catching .. 5.ish to 1
Analytical skills are in big demand so it is really important not to make the basic, common, mistakes that show you up as a newbie. For example, probability calculations are often performed on binary outcomes such as "What is the probability that a given policy holder will claim?" The result is binary because they will either …
The 'correct' calculation has not factored in the test frequency whatsoever.
If the tests were run on 52 consecutive days, if the drunk managed to miss out on being caught for those his chances for the rest of the year are 100%, even if he comes in drunk on 12 consecutive days.
Also, human behaviour being what it is, isn't it more likely that it will be a Monday that the person would be drunk, or a Friday afternoon for example?
Still, useful article :P
Yes, it has the test *frequency* in: on average once per month or once per year or whatever. It says nothing about how these should be distributed over the year, so the assumption is randomly. This is what frequency means in stats. This is what those people we call "frequentists" strive on.
What you describe is factoring in prior knowledge, and that is something that Bayes' rule can do for you - but not within the scope of that article... (I leave looking up a relevant xkcd as an exercise for the reader).
Mine's the one with the hip flask...
If tests were on Friday mornings you may stand a better chance of catching people as they start to let their hair down Friday nights. This would further interfere with the probabilities.
Theres a brilliant book on the history of stats - let me find it . . . blah, bloody Google - its on my Kindle - 2 mins . . . Against the Gods: The Remarkable Story of Risk. Very highly recommended.
Sheesh, you guys do love to overcomplicate these things...
Testing happens 1 day in 5 ("weekly"). So each time the target comes in drunk, they've got a 20% chance of being tested that day.
If they do it 12 times a year, their chance of getting away with it every time is (0.8 ^ 12 = ) 6.87%.
I thought in the real world you only tested people you wanted rid of anyway, in the hope they'd show up drunk and make it easy for you?
Probably less the case with Network Rail (who really don't like there poor employees drinking), but I've heard of it working that way in the private sector, especially banks and the like.
"I thought in the real world you only tested people you wanted rid of anyway, in the hope they'd show up drunk and make it easy for you?"
One of the customers I visit from time to time will grab anyone who is on-site, employee or visitor, for random drug/alcohol testing - it's something you have to agree to to be allowed on-site at all.
>Chance of catching a tea teetotaller in a year is 0%
Not here stateside. Gave up booze when hit my 30s as well as cigs. Now I am old, almost live like a Mormon minus consuming enough caffeine in a day to kill a horse. Have to feed the addiction side of my brain somehow and one of the safer ways I suppose.
and depending upon your physiology, you can fail a drugs test (morphine) from eating two Poppy Seed covered toasted bagels in the morning.
Don't worry people, your Spy TV/Echo/Bixby in the corner will ensure that your Employer (aka Big Bro) knows how much you are eating and drinking when you are at home. They will make sure that you are tested before you leave home for work. Can't have any liability falling on the Dear Employer now can we?
This, you understand, is a purely interim measure. You job will be replaced by robots/A.I. within 5 years. Then you won't be needed at all.
.... we get tested EVERY day. - plus for-cause and incident investigations. Not to mention random saliva and urine drug testing from both employer and client.
The system to not get caught is to not to consume - or know how much you can consume and metabolise safely.
At a previous workplace, where they were struggling to keep hold of employees, testing was carried out late in the day, a week into the swing and with the intent of not catching anyone.
@The Man Who Fell To Earth
Can't comment with any accuracy on non-working practices in Italy or Spain but you are dead right about France. I was employed in France for 30 years, only went in to the office for the jokes and the croissants, salary ok and to top it off I'm now retired on a decent pension*.
The joke is on you.
* My UK pension is spent in the UK, it provides pocket money for holidays.
Nah it's typically 10 holidays and then 2-4 weeks of PTO, sometimes split up into vacation/sick. Except for the companies that provide "unlimited" vacation. At my current job, after totaling up holidays, vacation, sick, personal days, and volunteer days, a work year is 225 days.
"The cumulative probability of getting away with it twice is 0.95 x 0.95 = 0.90 (actually 0.91 if you use greater precision)."
At two significant figures of precision, it's either 0.90 or 0.91 You could claim 0.9 and 0.91 to different levels of precision, but when you show that trailing 0, you are showing that you are being precise to that level.
Besides: 0.95*0.95 = 0.9025 which doesn't round to 0.91 at any precision.
If you use (248/260)*(247/260) for full precision, that comes out as 0.906154 which may be where you get your 0.91 from, but the article doesn't make that clear.
"Besides: 0.95*0.95 = 0.9025 which doesn't round to 0.91 at any precision.
If you use (248/260)*(247/260) for full precision, that comes out as 0.906154 which may be where you get your 0.91 from, but the article doesn't make that clear."
" which doesn't round to 0.91 at any precision"
So does it, or not
I'm confused by your comment.
Firstly, your calculation looks wrong to me it should be (248/260) * (247/259) which comes out to 0.909652 which rounded to two decimal places is indeed 0.91
Lastly, when I read the author saying it was 0.91 with greater precision I naturally assumed he meant with all calculations up to that point using greater precision. I don't know how else you could read it since he obviously isn't adding extra precision to the result of 0.95 * 0.95.
I work in an office that, among other things, is currently doing civil engineering design work for a rail project.
Train drivers must be dry, obviously, but there are no train drivers on the design team, and none of us will ever go near a train as anything other than a passenger. However, everyone working on the project must be dry, as a condition of the contract.
Rail projects are funny.
"Train drivers must be dry, obviously, but there are no train drivers on the design team, and none of us will ever go near a train as anything other than a passenger. However, everyone working on the project must be dry, as a condition of the contract."
Yep. It's been that way since 1991. There was a fatal buffer stop crash at Cannon Street caused by the driver not braking properly. The driver tested positive for cannabis three days later, though it was impossible to determine whether and if so to what extent that might have contributed to the crash. Nevertheless the Chairman of British Rail, Bob Reid II, brought in an immediate ban on the use of drugs and other substances which could impair concentration, including alcohol. Obviously the new rules could have just been applied to drivers and other safety critical workers such as signallers, but Reid thought it wasn't fair to insist that drivers couldn't have a lunchtime pint whilst managers and executives, including himself, still could. So the entire organisation went dry, overnight, right from the very top down. And after British Rail was broken up by the privatisation process, all its successor companies inherited the same policy.
"It assumes [..] that an employee is coming in with alcohol in their system (drunk) 12 times in a working year."
If you have one glass of wine with your noon meal, you do have alcohol in your system. You are not necessarily drunk, though.
What this means is that it is apparently considered that 0% is the only acceptable percentage of alcohol in the blood in a working environment. Seems a tad draconian to me, but I'm French, so . .
On the other hand, any level of alcohol when coming in to work in the morning is obviously unacceptable because it means that the person has been drinking on the way to work, which is a clear sign that they need professional help and cannot function normally.
The way I heard it, if you have more than four pints of an evening, you're risking being over the drink drive limit in the morning.
So if you have four pints the night before, you will have alcohol in your system, just not enough to lose your driving license over.
( Whether that four pints is actually correct I'm not sure, but you get the picture )
A lot of beers are significantly more than 2 units/pint. Looking at a couple of bottles conveniently to hand, Glamorgan Welsh Pale at 4.3% is 2.2 units (for 500ml, so about 2.4 for a pint), and Three Tuns Old Scrooge at 6.5% is 3.3 (3.6 for a pint).
But serious kudos (and concern for your liver) if you can manage 4 pints of Old Scrooge in an evening! You'll still be showing the effects at lunchtime (if you make it out of bed)
Correct, kind of. But that 8 hours assumes you drank all 4 pints at once.
units of alcohol is about strength % * volume (in litres)
So a metric pint (half litre) of 4% beer is 2 units.
If you begin at at fairly normal 20:00, drink 4 pints between then and 00:00, that's already 2 of the pints gone from the system, the next 2 by 02:00. Driving to work the next day at 08:00 won't be an issue.
(This is a rough estimate as it takes some time for your liver to start processing the alcohol obv.)
In the UK, the drink driving limit, roughly, for a 12 stone man, is around 4 units, so you would unlikely be over the limit if you waited an hour or so after midnight and then drove (but I wouldn't recommend risking it).
Wasn't aware that beer could have that kind of effect.
So I have to amend my position on alcohol in the morning, since it could simply mean that you had "too much" the night before. I'm okay with that, although I will be encouraging you to drink lots of water during the day.
The quick way is to do 1-(0.95)^52=0.93 (2 sig figs), which gets you almost exactly the right answer. That's how we were taught to do it in school for continuous events and adding probabilities. The extra accuracy you get by withdrawing balls gives you the correct answer, but the short form above works well, if the testing is still relatively infrequently.
> The quick way is to do 1-(0.95)^52=0.93 (2 sig figs),
> which gets you almost exactly the right answer.
No, it gets you exactly the right answer. The use of a hypergeometric distribution is technically correct (but only if the sot keeps track of how many times he's been drunk and stops after twelve such days); personally, I would apply a binomial distribution with a fixed probability of drunkenness on any one day of p=12/260, just as you did.
The results are: P(X>=1) = 0.935646978954 for the hypergeometric case, P(X>=1) = 0.914321213168 for the binomial distribution. The commenter that decided that the result must round to 0.90 is living in a state of sin.
"No, it gets you exactly the right answer."
It gets you the right answer, but to a different question. It gets you the right answer to the question "what is the chance of getting caught given that each day has a fixed probability of you being drunk", whereas the question was that in a given year there are exactly twelve drunk days, so it becomes a finite problem which needs finite methods rather than a continuous probability.
I think your rough calculation is actually closer to correct, while the one given in the article is slightly wrong. This is not a case of "removing one ball from the bag" each time as the writer has assumed. In actuality it is likely that the probability of the employee being drunk on any given day is always 12/260, independent of any previous day. In other words, the employee decides on any given day "do I need a drink today?" depending on his or her current circumstances, and is not thinking "wait, I've already gone in to work drunk 5 times this year, so maybe not today.".
I think we accidentally get hung up on the pertinent request, "If I do something X times, what are the facts about X?". In this case we get caught up on getting to that magic fact being driven by the 52 times/year, the 4 times/year or twice a year and start from there. Just human nature I suppose to look for the quickest route through a problem rather than the correct logical route.
The "bag of marbles" visual idea for probabilities takes me back to my school days and later on working through maths homework with my own kids. It's such a great little visual image that immediately registers with the imagination. Don't try to imagine an abstract idea of 12 days over 260 because you cannot be two mutually exclusive things at the same time ( drunk and sober ) , but imagine it's ( 260 = 12 + 248 ) , 12 red marbles and 248 green marbles. Immediately most people, even those who really struggle with maths, can visualise instantly and start to get a handle on probabilities. Oddly my imagined bag of marbles is always dark grey felt with a black draw cord, ha ha!!
I've long tended to grab two socks from the drawer in the morning without really checking to see if they match. SWMBO objected to this behavior. I caved in (always a good idea on minor issues; choose one's battles carefully, etc.) and have since taken a moment or two to ensure a match. But I did explain to her that my previous behavior was just in memory of (seemingly countless) probability problems as a lad such as:
Eddie has five red socks, three black socks, and two white socks. If Eddie grabs two socks in the morning at random, what's the probability that he'll grab two socks that match?
If one of my employees shows up obviously under the influence, I'll show them the door. It's that simple, and they know it. They have the option of taking a blood test immediately after being fired, and I'll deal with the consequences if I'm wrong. Thankfully I have never needed to resort to this ... But that's not to say I'll fire 'em if they have booze on their breath. I operate a small brewery, winery, and fledgling distillery. Almost all of my employees have access to, and in some cases are required to sample the product ... Either you trust your employees, or you don't. If you don't, get rid of them. Makes life much easier all around.
Buzzword, I'm fairly certain I can tell the difference. My velcro-whippet certainly can. I'll make the call if/when. And as I said, if I'm wrong I'll face the consequences.
Hasn't happened yet. Somehow, I doubt it will ever be an issue. Not with this crew; they've all been with me for nearly two decades, and all have a piece of the holding company.
Can you tell the difference between someone who is still drunk from the night before, and someone who has a cold? What if they simply haven't slept properly because of external factors (heat wave, noisy neighbours, etc.)?
Yes, because of the smell of ethanol coming from their skin pores.
"If one of my employees shows up obviously under the influence, I'll show them the door. It's that simple, and they know it. They have the option of taking a blood test immediately after being fired, and I'll deal with the consequences if I'm wrong."
If you are in the lovely US in one of those at will states then fine, but in the UK you would be in court pretty quickly with that kind of approach.
Instant dismissal isn't really necessary. When coming back from a liquid lunch, and having been judged to have forgotten how to count above two pints, my manager would meet me at the office door and offer me a pen and an instant, one day, holiday request form. Then off home. It was usually worth it. Fighting in the car park optional.
I've been at places where when we know we are doing a lunchtime celebration then the network access is revoked in advance. I've also been on lunch celebrations where the boss decided it would be better for people to go straight home from the pub rather than be let loose on the IT systems.
How many people do you have to have in a room before there is a greater than 1 in 2 chance that 2 of them have the same birthday?
People tend to guess a number around 183, but the same logic as the drunk testing applies and the answer is in the low 20s. (It is not exactly as calculated because there tend to be more births at certain times of year.)
Hmm. Overthinking it a bit.
My one and only experience of a drunk at work goes back to my first job in the early 90s.
The 'drink because my wife and kids have left me because I moved the family from <somewhere nice> to Bracknell' rolled in at 3pm, fell asleep, then stood up and peed in his top desk drawer.
Sitewide no boozing at work policy the next week.
"The 'drink because my wife and kids have left me because I moved the family from <somewhere nice> to Bracknell'"
Bracknell = RACAL perchance.
"rolled in at 3pm, fell asleep, then stood up and peed in his top desk drawer." = https://www.youtube.com/watch?v=oNFz0hkPXvA
There was a similar sort of Christmas party at RACAL Seaton (Before I joined), there was sex under benches & all sorts of celebrations on return from the pub\in house supplied booze, following year Management wizened up & no such celebrations took place, they just kicked us out at lunchtime & we went straight to The George instead.
We know they test once a week. Wait sober until they test, and drink until the weekend. Probability of being caught 0!
Alternatively, it's almost certain that the boss will have been seduced by sales spiel into buying the latest fanciest internet connected alcohol testing machine. Even the greenest PFY should be able to insert the 'if dept = IT then alcohol = random number between totally sober and reasonably sober' into the code. Or maybe the simpler 'if dept = HR then alcohol = embarrassingly pissed'
And pedantically, assuming that the employee's drinking is reasonably spread your green and red balls are a bit skewed - (I think - of course last night's bender might be clouding my judgement), because while in week 1 you may have 248 red 12 green, but week 2 we should be calculating 243 red 11.75 green, - the odds of being caught aren't changing any more than the odds of throwing a coin change? Your numbers assume that if I was sober on test 1 the chances of me being sober on test 2 are slightly less, effectively assuming that there are still 12 drunk balls, but now 247 sober ones.
I think it was just a poorly labelled table and the best way of thinking about it is there's a 12/260 chance that I'm drunk on a given day and a 52/260 chance that I will be tested on a given day
so for a given day the probability of getting away with it = 0.04 * 0.2 = 0.009 as it was in the table
They just forgot that I come to work 260 days a year (for which I'd need to be on something) so
260*0.009 = 2.4, a 240% chance of being busted. That empirically feels right, If I do something where i have a 20% chance of being caught 12 times in a row, I'm going to get caught.
Define "drunk" :)
Back in the early 90's a company I worked for had a policy. The actual wording was "you must not attend work drunk to the extent of being incapable". Left quite a lot of scope for a pint or three at lunchtime (and to be fair there was no risk of working with alcohol in the system other than not getting the work done).
Indeed. Assuming testing first thing in the morning (before you drive your train/bus/lorry into people), you will merely ensure not being too tipsy for these activities at that time.
Which leads to inevitable lunchtime drinks, and afternoon drinks, and then getting home and not drinking, because you've got to get your liver to process six pints in the next twelve hours.
This is actually a far better way to do drinking than the British 'neck a load in the evening until 1am' methodology, that guarantees a nasty hangover the next day.
So taking the bag of marbles example, if I'm being tested once a week then for the first test there are 248 green marbles and 12 red marbles, (total 260), but on the 2nd test there will be 5 days (assumign 5 working day week) less surely - not 1x less? (and dont get me started on how we work out how many green/how many red we need to take out)
"but on the 2nd test there will be 5 days (assumign 5 working day week) less surely - not 1x less? (and dont get me started on how we work out how many green/how many red we need to take out)"
I think I convinced myself that summing over all possibilities for removing n balls at random for varying n gives you the same answer as if you just remove a single green ball. I didn't prove it, but that's because I couldn't be bothered.
I'm surprised it took till page 3 for me to find someone else saying this, specifically, how do you know (apart from on day 1) how many red balls are in there?
Every day, someone takes out 1 ball, but you don't know what colour it is. so there is a possibility that after say, 15 days, all 12 red have been taken out, but because they were removed on non test days, you have no idea. As the tester, every day (whether you are testing or not) there are between 0-12 red balls in the bag.
As others have said, there is a chance that people are coming in drunk on consecutive days, so that could be two or three reds in a row, which would increase the immediate chance of getting caught, but in the long term it would decrease it (which i guess balances out?)
i think I'm getting carried away here because it's Friday morning and I'm full of caffeine, but, with the 'balls test' (in this case) surely the correct way to conduct it, is for two people to be involved, one (the 'drunk', chooses a ball from the bag, and keeps it to them selves, unless the other person (the 'tester') asks them what it is, repeat until either the bag is empty or the 'drunk' gets caught.
The probability of me being drunk tonight after I finish my fear, frustration, disgust, and despair-inducing day at work today approaches 100%. (I also have a funeral to go to today)
At work? Nah. I like to be sober during the day just to mix things up a bit. Unless you count our company picnic..
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I was recently faced with the prospect of signing a contract that insisted I take part in the company's testing policy, even though I work for myself.
No amount of negotiating got them to budge, and I was on the verge of rejecting the contract (out of principle you understand) when I decided to read the full policy in detail.
Turns out the selection process is based on
2. A list of the people signed in using their access card that morning at that location
3. A random selection of the people from the list.
Since I work remotely, and when I do go to the main office I have to sign in as a guest (I have no access card) there is no chance of me appearing on the list in #2, so probability of being tested = 0.
So I signed the contract :)
So....If I wait until I've been tested and then get very, very, drunk, what are the chances of me being caught the next day??
If it helps your calculations:
1: I'm drunk now. 3 pints of 4.5% real ale at lunchtime.
2: Yes I do appear to be a lightweight but I could drink a lot more if I wanted.
3: But only if I eat more than a piece of toast for breakfast.
4: I think I'll have pie and chips for tea.
5: That's northern tea at 6pm not the confused southern 'tea' which is actually dinner. Or lunch. Brunch?
6: What was the question again?
7: Hang on ! I don't even have a job! I could be rat-arsed anytime I want!!!
We cannot know what logic allowed the producers of the table to arrive at the incorrect value of 0.9 per cent
Here's the logic: the producers of the table are marketers who know fuck all about probability but plenty about how to persuade companies to sign up for the highly-profitable daily testing regime.
I ran it using my calculations and got the following:
Test once per year - 4.62%
Test twice per year - 9.02%
Test every month - 43.28%
Test every week - 91.43%
Test every day - 99.9995%
The assumption I made is that every day, there is a 12/260 probability that they will go in drunk, and that each day, the probability of going in drunk is not determined by how many times they have gone in drunk so far in the year, so there is a very small chance that they might go a whole year without getting drunk.
"The assumption I made is that every day, there is a 12/260 probability that they will go in drunk, and that each day, the probability of going in drunk is not determined by how many times they have gone in drunk so far in the year, so there is a very small chance that they might go a whole year without getting drunk."
This has been answered above, it's the difference between continuous probability (as you used) and discrete probability, which is what the assumptions given in the article.
You are also correct that continuous is, for most practical purposes, a better model. But you pick your model based on other needs. The answers are also pretty much the same, apart from daily being 1 rather than almost 1 :)
The assumptions are that the worker is drunk, is testable as drunk, etc for exactly 12 out of 260 days each year. If they have been drunk at work for 12 days, they stop turning up at work drunk.
I *think* the more logical explanation for the numbers is that a couple of marketers have had a whack at it.
So the *method* for the numbers I believe is
1. Correct finite probabilty calculations have been done, as in the article.
2. It's put in a nice table.
3. It's easy to understand, but it doesn't really sell the product anyway more than common sense will
4. This is bad. Lets edit those numbers
5. OK, first off, bollocks to this 0-1 range. Percentage. 1 = 100% = the best. That goes in first
6. Working backwards, 0.94, that rounds to 0.9, whack a percentage in front and we're done. Wait, percentage at the back. 0.94 = 0.9%. Good job my bonus math is better huhuhhuh.
7. OK, 0.43 that rounds to 0.4, thus 0.4%. Bit high, boss wants to push the weekly test kits. Halve it. 0.43 = 0.2%
8. Allright, lets not reinvent the wheel. halve, round down and percentagise those last buggers.
There we go, conveys the correct "sales story" and explains why anyone could come up with those numbers. I feel the author was too charitable in their assessment of where the "figures" came from.
It's data so badly faked it wouldn't fool an accountant. It's almost funny how bad it is at actually selling the product. One hopes that someone got paid a lot of money to say "multiply by 100, put a percentage at the end", in an expensive advertisement for paying attention at school.
Flicker fusion challenge via random phone selection (at 1+ hour from start time).
Non discriminatory, if someone is tired, drunk, has severe 'flu etc they are politely asked not come in until better.
Also helps reduce incidences of people crashing their cars due to working three jobs or ridiculous hours (it happens!) and everyone benefits.
I did wonder if the adding of IR thermometers to phones might reduce seasonal illnesses as a temperature is often an early sign of influenza and Norwalk aka winter vomiting bug.
Are we assuming that the weekly test is the same day of the week, or evenly distributed throughout the year?
Does the person come in drunk on a Monday only or are they an alcoholic that comes in drunk 4 days a week (ie Mon-Thu) 3 times a year? If the testing is every Friday neither will get caught.
There's not enough data to give an accurate result.
Says the guy who has just drunk two glasses of prosecco at work (with permission!) and it's gone to his head.
Quite a few of us used to go in to work quite drunk (we had a a bar on site), straight after lunch (did I mention the bar on site? (50p per pint!!!)). Alas it got to a point (not surprisingly), that some (myself included) were far beyond drunk and into holy shit, he's completely pissed!!! territory (work from home now, and pace myself (I'm an old git now)).
Not something I'm proud of, but testing wouldn't have helped (I did mention the bar on site, right? (disappeared not long after myself and a few others were fired for being shitfaced after lunch, from what I heard (not surprisingly))), limiting how much the bar would sell you, or a strict "look you can have have this much as the on-site or off-site bar", would've) - yes, I was young and stupid - weren't we all?
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