# Explaining Drug Testing

#### S-Scitation

##### New member
Hello All, am looking for some help explaining probability to drug test participants.
Basis 250 Employee, 50% of population sampling per year criteria approximately 11 selections per month
Trying to explain that 11 opportunities/month to be selected out of 250 employees is not 11/250 but rather 1/250,1/249, 1/248,1/247......1/241 etc. What is percent probability to be selected each month assuming name replacement? Looking for easy explanation to nervous people. Help appreciated

#### tkhunny

##### Moderator
Staff member
1) If employees are that worried about a drug test, you have bigger problems.
2) Maybe 11/250, 10/249, 9/248, etc. I assume you draw 11 without replacement, and then put all 11 back after you're done.
3) There is no point to drug testing if you put people's names on a calendar. There's a reason it's called "random". Do you REALLY want someone who just passed the drug test to think it is now clear and there will be no testing for the next year? That's no good.
4) Everyone in the pot. Anyone can be selected - every time. ANY other method is pointless.
5) It's possible there are federal guidelines that disagree with my assessment, but federal guidelines don't always make sense.
6) What do you do with a failure? Throw them back in the random population or make sure they do get tested next month? Would this count as one of your 11?
7) I can also see that racial mix or sex could be a problem. For example, if you have 248 men and 2 women, representatives of the men may object that the women never get tested. Of course, if you ALWAYS select one of the 2 women, they will soon cry foul! No easy solution for this problem.

#### Dr.Peterson

##### Elite Member
Basis 250 Employee, 50% of population sampling per year criteria approximately 11 selections per month
... What is percent probability to be selected each month assuming name replacement?
To directly answer your central question, it is entirely true that, if 11 people out of 250 are selected each month, then each person has a probability of 11/250 = 4.4% of being selected in a given month. Each person therefore (assuming each month's selections are random and not affected by previous months) has a probability of 1 - (1- 0.044)^12 = 41.7% of being selected at least once in a given year.

I presume your mention of 50% per year means that the number 11 was chosen so that about half the people are tested over a year, by finding (0.50*250)/12 = 10.4. The annual probability above, 41.7%, is less than 50% largely because of the possibility of being selected more than once. But clearly it is by design that something close to half of the employees are selected in a given year, so you might as well be honest about it.

As tk said, anxiety about this suggests there is more going on; and rather than hide the truth, you might want to deal with the underlying issues. I would think (not as an expert on this) that it might be better to explain why the testing is (a) required, or (b) good for the company, or (c) not likely to result in false positives, or whatever is true and meaningful to them. I can imagine many reasons to be nervous about it other than guilt, but you might want to find out about the actual reasons.

#### S-Scitation

##### New member
To directly answer your central question, it is entirely true that, if 11 people out of 250 are selected each month, then each person has a probability of 11/250 = 4.4% of being selected in a given month. Each person therefore (assuming each month's selections are random and not affected by previous months) has a probability of 1 - (1- 0.044)^12 = 41.7% of being selected at least once in a given year.

I presume your mention of 50% per year means that the number 11 was chosen so that about half the people are tested over a year, by finding (0.50*250)/12 = 10.4. The annual probability above, 41.7%, is less than 50% largely because of the possibility of being selected more than once. But clearly it is by design that something close to half of the employees are selected in a given year, so you might as well be honest about it.

As tk said, anxiety about this suggests there is more going on; and rather than hide the truth, you might want to deal with the underlying issues. I would think (not as an expert on this) that it might be better to explain why the testing is (a) required, or (b) good for the company, or (c) not likely to result in false positives, or whatever is true and meaningful to them. I can imagine many reasons to be nervous about it other than guilt, but you might want to find out about the actual reasons.
Thanks for the response much appreciated. You are correct in the set-up 50% arriving at 11 selections per month. I am looking for a way to present short and simple odds to eliminate all the subjective opinions but more importantly truly educate the workforce to eliminate fear that the process is not trying to be fair. The method and program is pretty sound, random number generator etc, validates with the DOT program, outside company makes the random selections etc. Employees are more anxious about fairness getting selected more than once during the year but the sampling is with replacement and everyone has an equal chance every month. The real concern seems to be how to present the odds of selection per month to the employees in a manner they understand. One example is one employee says you pick 10 names per month so I have 10 chances of being picked out of 250 so my odds are 10/250 1: 25 but really isn't their chance 1/250 followed by 1/249 in succession until the 11th pick is produced. Those that get picked more than one month in a row or 3 times per year think they have satisfied some kind of amazing odds for that happening while others complain that they have never been picked. As an example in roughly 11 months there were 130 selections, 19 people randomly being selected more than once represented 43 selections out of the 130 selections. One employee was selected 4 times, three employees selected three times and 15 employees were selected twice. While one might say that does not look random with repeat selections another would offer that with the random number generator that is possible and is demonstrating randomness, others might say it looks like the selection process and random number generator are stuck.

#### tkhunny

##### Moderator
Staff member
Thanks for the response much appreciated. You are correct in the set-up 50% arriving at 11 selections per month. I am looking for a way to present short and simple odds to eliminate all the subjective opinions but more importantly truly educate the workforce to eliminate fear that the process is not trying to be fair. The method and program is pretty sound, random number generator etc, validates with the DOT program, outside company makes the random selections etc. Employees are more anxious about fairness getting selected more than once during the year but the sampling is with replacement and everyone has an equal chance every month. The real concern seems to be how to present the odds of selection per month to the employees in a manner they understand. One example is one employee says you pick 10 names per month so I have 10 chances of being picked out of 250 so my odds are 10/250 1: 25 but really isn't their chance 1/250 followed by 1/249 in succession until the 11th pick is produced. Those that get picked more than one month in a row or 3 times per year think they have satisfied some kind of amazing odds for that happening while others complain that they have never been picked. As an example in roughly 11 months there were 130 selections, 19 people randomly being selected more than once represented 43 selections out of the 130 selections. One employee was selected 4 times, three employees selected three times and 15 employees were selected twice. While one might say that does not look random with repeat selections another would offer that with the random number generator that is possible and is demonstrating randomness, others might say it looks like the selection process and random number generator are stuck.
One more thing. Your 50% assumption assumes NO repeats. If you really want a 50% expectation, you'll have to move it up.

#### JeffM

##### Elite Member
If you are going to test 11 different individuals each month out of a population of 250, the probability that any one will be selected in a specific month is indeed going to be 11/250 = 4.4%. The probability that a given individual will be picked first is 1/250. The probability that individual will not be picked first is 249/250. So the probability that individual is picked second is zero if the that individual was picked first and is 1/249 if that individual was not picked first. We are dealing with conditional probabilities. So to calculate the probability that individual is picked second is (0)*(1/250) + (1/249)(249/250) = 1/250. And that gets repeated each time.

$$\displaystyle \dfrac{1}{250} + \dfrac{249}{250} * \dfrac{1}{249} + \dfrac{248}{250} * \dfrac{1}{248}$$

$$\displaystyle \dfrac{247}{250} * \dfrac{1}{247} + \dfrac{246}{250} * \dfrac{1}{246} +$$

$$\displaystyle \dfrac{245}{250} * \dfrac{1}{245} + \dfrac{244}{250} * \dfrac{1}{244} +$$

$$\displaystyle \dfrac{243}{250} * \dfrac{1}{243} + \dfrac{242}{250} * \dfrac{1}{242} +$$

$$\displaystyle \dfrac{241}{250} * \dfrac{1}{241} + \dfrac{240}{250} * \dfrac{1}{240} =$$

$$\displaystyle \dfrac{1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1}{250} =$$

$$\displaystyle \dfrac{11}{250} = 4.4\%.$$

As for explaining this, I'd just say that the probability of being tested in any given month is 4.4% and that an individual may be picked more than one time a year. Why try to explain probability theory, which many people find HIGHLY unintuitive.

#### Dr.Peterson

##### Elite Member
Thanks for the response much appreciated. You are correct in the set-up 50% arriving at 11 selections per month. I am looking for a way to present short and simple odds to eliminate all the subjective opinions but more importantly truly educate the workforce to eliminate fear that the process is not trying to be fair. The method and program is pretty sound, random number generator etc, validates with the DOT program, outside company makes the random selections etc. Employees are more anxious about fairness getting selected more than once during the year but the sampling is with replacement and everyone has an equal chance every month. The real concern seems to be how to present the odds of selection per month to the employees in a manner they understand. One example is one employee says you pick 10 names per month so I have 10 chances of being picked out of 250 so my odds are 10/250 1: 25 but really isn't their chance 1/250 followed by 1/249 in succession until the 11th pick is produced. Those that get picked more than one month in a row or 3 times per year think they have satisfied some kind of amazing odds for that happening while others complain that they have never been picked. As an example in roughly 11 months there were 130 selections, 19 people randomly being selected more than once represented 43 selections out of the 130 selections. One employee was selected 4 times, three employees selected three times and 15 employees were selected twice. While one might say that does not look random with repeat selections another would offer that with the random number generator that is possible and is demonstrating randomness, others might say it looks like the selection process and random number generator are stuck.
Thanks; that gives a much better sense of what the issues are. I tried to look at what DOT rules you might be referring to, and I don't want to even try read through it all! (If it does say somewhere that at least 50% must be tested each year, I'd want to see the exact wording of the rule, to see what it would take to comply.)

So the issues are fairness (both a priori, and given evidence of multiple selections or not being selected in practice), and mere understanding of the probabilities.

First, as I've said, the probability of being chosen in a given month is 11/250 if 11 are picked. The numbers 1/250, 1/249, ..., are the probabilities of being the first selected, or the second (given you weren't the first), and so on. It turns out that if you add up the probabilities of being chosen first, second, ..., eleventh, the total works out to 11/250, and that's all that counts. But I don't think the actual number is the real issue for most people.

Second, as at least one person evidently is aware, random selection implies more repetition than we commonly expect. If no one were selected more than once, then I would be questioning the randomness of the selection! So one thing to do is to analyze the observations and see whether it is compatible with random selection. (But that would be far too technical to impress most people, unless they just like to hear some independent authority say it.)

Would you like someone to compare the data you provided with expected numbers picked 0, 1, 2, 3, 4 times? Or is that not worth the effort?

#### S-Scitation

##### New member
Thanks; that gives a much better sense of what the issues are. I tried to look at what DOT rules you might be referring to, and I don't want to even try read through it all! (If it does say somewhere that at least 50% must be tested each year, I'd want to see the exact wording of the rule, to see what it would take to comply.)

So the issues are fairness (both a priori, and given evidence of multiple selections or not being selected in practice), and mere understanding of the probabilities.

First, as I've said, the probability of being chosen in a given month is 11/250 if 11 are picked. The numbers 1/250, 1/249, ..., are the probabilities of being the first selected, or the second (given you weren't the first), and so on. It turns out that if you add up the probabilities of being chosen first, second, ..., eleventh, the total works out to 11/250, and that's all that counts. But I don't think the actual number is the real issue for most people.

Second, as at least one person evidently is aware, random selection implies more repetition than we commonly expect. If no one were selected more than once, then I would be questioning the randomness of the selection! So one thing to do is to analyze the observations and see whether it is compatible with random selection. (But that would be far too technical to impress most people, unless they just like to hear some independent authority say it.)

Would you like someone to compare the data you provided with expected numbers picked 0, 1, 2, 3, 4 times? Or is that not worth the effort?
Again much appreciated from all. Explanations are awesome. I like the pure math and I truly believe taking some time to explain fairness to employees produces a much more productive and educated workforce so it is worth the effort. People feel comfort when they know you are at least trying to make sure they fully understand the process. The attached chart can help explain the probability of selection to them. The help with the math portion of explanation is also to be used. I was wondering how to go about the math probability to produce a chart showing the probability of being selected 2, 3, 4 or more, using the same chart parameters. I have also been asked on options to change from 50% selection 12 months to have a better opportunity of reducing the repeat selections without changing the random process selection. Your thoughts ?

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#### tkhunny

##### Moderator
Staff member
Does anyone really care if they are picked first or eleventh?

#### Dr.Peterson

##### Elite Member
Those that get picked more than one month in a row or 3 times per year think they have satisfied some kind of amazing odds for that happening while others complain that they have never been picked. As an example in roughly 11 months there were 130 selections, 19 people randomly being selected more than once represented 43 selections out of the 130 selections. One employee was selected 4 times, three employees selected three times and 15 employees were selected twice. While one might say that does not look random with repeat selections another would offer that with the random number generator that is possible and is demonstrating randomness, others might say it looks like the selection process and random number generator are stuck.
The attached chart can help explain the probability of selection to them. The help with the math portion of explanation is also to be used. I was wondering how to go about the math probability to produce a chart showing the probability of being selected 2, 3, 4 or more, using the same chart parameters. I have also been asked on options to change from 50% selection 12 months to have a better opportunity of reducing the repeat selections without changing the random process selection. Your thoughts ?
I've made a spreadsheet to match your table (which doesn't take into account rounding to a whole number of people each time, but is close to the numbers we got before), and modified it to list probabilities of being selected exactly 0, 1, 2, 3, 4 times, or at least 1, 2, 3, 4 times. Of more interest, perhaps, at the moment, are the expected numbers for something like your specific conditions. Here are my results:

This is pretty close to the numbers you reported, so it actually does look quite random. (The numbers shown are the 250 times the probability of each event.)

Here is the formula in cell B6: =B$1*COMBIN(B$3,$A6)*(B$2/B$3)^$A6*(1-B$2/B$3)^(B$3-$A6)

This could also be done using the Binomial Distribution function: =B$1*BINOM.DIST($A6,B$3,B$2/B$3,0). But you like the math, ... . As for changes: if you decrease the probability of being selected more than once, you will not be doing random selection. Indeed, if you eliminated the possibility of being selected more than once, you would be violating the point of the testing, as those people would know they are "safe". #### S-Scitation ##### New member Does anyone really care if they are picked first or eleventh? I would say there is not as much concern from employees about the place number of selection(1-11), but rather that places in a list do exist, they can be seen and that if selected more than once that the place is not always the same place. #### S-Scitation ##### New member I've made a spreadsheet to match your table (which doesn't take into account rounding to a whole number of people each time, but is close to the numbers we got before), and modified it to list probabilities of being selected exactly 0, 1, 2, 3, 4 times, or at least 1, 2, 3, 4 times. Of more interest, perhaps, at the moment, are the expected numbers for something like your specific conditions. Here are my results: View attachment 12567 This is pretty close to the numbers you reported, so it actually does look quite random. (The numbers shown are the 250 times the probability of each event.) Here is the formula in cell B6: =B$1*COMBIN(B$3,$A6)*(B$2/B$3)^$A6*(1-B$2/B$3)^(B$3-$A6) This could also be done using the Binomial Distribution function: =B$1*BINOM.DIST($A6,B$3,B$2/B$3,0). But you like the math, ... .

As for changes: if you decrease the probability of being selected more than once, you will not be doing random selection. Indeed, if you eliminated the possibility of being selected more than once, you would be violating the point of the testing, as those people would know they are "safe".
Dr. Petersen, my sincere thanks for your assistance. and for some personal peace at mind at my daily task assignments. Employees do have a better sense of trust on a sensitive subject when informed. Thank you again.

#### S-Scitation

##### New member
If you are going to test 11 different individuals each month out of a population of 250, the probability that any one will be selected in a specific month is indeed going to be 11/250 = 4.4%. The probability that a given individual will be picked first is 1/250. The probability that individual will not be picked first is 249/250. So the probability that individual is picked second is zero if the that individual was picked first and is 1/249 if that individual was not picked first. We are dealing with conditional probabilities. So to calculate the probability that individual is picked second is (0)*(1/250) + (1/249)(249/250) = 1/250. And that gets repeated each time.

$$\displaystyle \dfrac{1}{250} + \dfrac{249}{250} * \dfrac{1}{249} + \dfrac{248}{250} * \dfrac{1}{248}$$

$$\displaystyle \dfrac{247}{250} * \dfrac{1}{247} + \dfrac{246}{250} * \dfrac{1}{246} +$$

$$\displaystyle \dfrac{245}{250} * \dfrac{1}{245} + \dfrac{244}{250} * \dfrac{1}{244} +$$

$$\displaystyle \dfrac{243}{250} * \dfrac{1}{243} + \dfrac{242}{250} * \dfrac{1}{242} +$$

$$\displaystyle \dfrac{241}{250} * \dfrac{1}{241} + \dfrac{240}{250} * \dfrac{1}{240} =$$

$$\displaystyle \dfrac{1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1}{250} =$$

$$\displaystyle \dfrac{11}{250} = 4.4\%.$$

As for explaining this, I'd just say that the probability of being tested in any given month is 4.4% and that an individual may be picked more than one time a year. Why try to explain probability theory, which many people find HIGHLY unintuitive.
Thank you Jeff for helping me with this I was a little confused at first with the addition sign but shortly realized the probability sums must be additive to 1. Thank you again.