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Probability Help

05/26/2021 10:33 AM

Attention geniuses and mathemagicians,

I need your help with some probability issues. Our school (my kids attend and I coach for) has a "random" drug screening policy for student athletes. For each testing period, 10 names are selected from a pool of approximately 175. Tests may be conducted monthly during the school year, so, perhaps 8 or 9 times per year.

I have heard testimony from at least 3 other parents that their children have been "randomly" tested MULTIPLE times (as many as 4-5) during the same school year. The testing administrator even admitted that this has occurred with some frequency.

He directed me to a website of the randomizer they use. The domain he gave me is for sale, as is the same domain name with a couple other common extensions I tried.

I realize that the above scenario is not impossible, but I also know that it is HIGHLY unlikely. When I asked the administrator "at what point do you question the integrity of your randomizer?" He had no reply.

So please, help me out here. At what point do you say, something fishy is going on? Please do supply any relevant statistical data you can come up with. It has been far too long since I sat through Statistics, so I could really use and would greatly appreciate your input!

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#1

Re: Probability Help

05/26/2021 1:11 PM

You have enough to say that yourself...

if the administrator gave you a website that is for sale... something is wrong where someone isn’t doing their job.

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#2

Re: Probability Help

05/26/2021 1:24 PM

Shooting from the hip, it looks like about a 50/50 chance that you will be selected during the year at least once...A particular student being chosen at random several times during the year is indeed probable...

https://www.indeed.com/career-advice/career-development/how-to-calculate-probability

https://www.dcode.fr/random-selection

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#5
In reply to #2

Re: Probability Help

05/26/2021 2:48 PM

It appear similar to getting picked for jury duty... again, and again, and again...

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#3

Re: Probability Help

05/26/2021 2:20 PM

Assume names are put back into the jar after each drawing.

For each child, the probability of getting drawn each time is p =(10/135) = .074074... and the probability of not being drawn is q = 1 - p = (125/135) = .925925...

If you have n drawings, the probability of being selected k times is:

P = (k!/(n-k)!)pkqn-k

https://en.wikipedia.org/wiki/Binomial_distribution

where k! is the product of integers 1 through k. (e.g. 3! = 1*2*3 = 6, and 7! = 1*2*3*4*5*6*7 = 5040).

Note: Many calculators have a factorial key "n!"

Example: if there are 10 drawings, the probability of getting selected 3 times is

P = (3!/7!)p3q7 = (6/5040)(.0741)3(.926)7 = 2.8233 x 10-7

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#4

Re: Probability Help

05/26/2021 2:42 PM

Thanks for the quick feedback. I'll have to plug some numbers when I have more time. Clearly, something VERY fishy is occurring.

Unfortunately, I cannot PROVE they are making false claims of randomization. I can, however, provide enough statistical evidence to hopefully make people draw their own conclusions. Either they are lying about the random nature of testing or they are far too incompetent to be trusted with administering these tests.

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#6
In reply to #4

Re: Probability Help

05/26/2021 3:19 PM

Oops, I have a typo that affects the numbers. My apology.

Here is the corrected post...

Assume names are put back into the jar after each drawing.

For each child, the probability of getting drawn each time is p =(10/135) = .074074... and the probability of not being drawn is q = 1 - p = (125/135) = .925925...

If you have n drawings, the probability of being selected k times is:

P = (n ! / (k ! (n-k) ! ) ) pk qn-k *****CORRECTION*****

https://en.wikipedia.org/wiki/Binomial_distribution

where k! is the product of integers 1 through k. (e.g. 3! = 1*2*3 = 6, and 7! = 1*2*3*4*5*6*7 = 5040).

Note: Many calculators have a factorial key "n!"

Example: if there are 10 drawings, the probability of getting selected 3 times is

P = (10 ! / (3 ! * 7 ! ) ) p3 q7 = (3628800 / (6*5040) )* (.0741)3 *(.926)7 = .0285.

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#11
In reply to #6

Re: Probability Help

05/26/2021 8:45 PM

I think you should calculate what the odds are for anybody getting picked three times over the course of 8 drawings....I ran this simulation 0ne time ...

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#16
In reply to #11

Re: Probability Help

05/27/2021 10:06 AM

Very good point.

First we need to calculate the probability of being selected 3 or more times for one child, and then calculate the probability for children out of the 175 that would have been selected 3 or more times. N = number of trials = 8.

If p=10/175 is the probability of being selected and q=165/175 the probability of not being selected on a given trial.

Expand (p+q)8 = p8 + 8p7q + 28p6q2 + 56p5q3 + 70p4q4 + 56p3q5 + 28p2q6 + 8pq7 + q8.

The first term is the odds of being selected all eight times, the second seven times, ... the last term not being selected at all. To figure for 3 or more times, just evaluate the last three terms (2 times, 1 time, 0 times) and subtract from 1.

I get P = 1 - (28p2q6 + 8pq7 + q8 ) = 8.4051e-03.

This is the probability of one child being selected 3 or more times.

The probability of one child not being selected 3 or more times is Q = 1 - 8.4051e-03 = 0.9916.

The probability of none of the 175 children being selected 3 or more times is Q175 = (0.9916)175 = 0.2283. The probability of 1 or more of the 175 children being selected 3 or more times is 1 - 0.2283 = 0.7717

Bottom line, there is a 77% chance at least one child of the 175 will have been selected 3 or more times.

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#14
In reply to #6

Re: Probability Help

05/27/2021 5:45 AM

Shouldn't that be p = 10/175?

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#25
In reply to #14

Re: Probability Help

05/27/2021 8:28 PM

Shouldn't that be p = 10/175?

Yup, I need to clean my glasses...

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#26
In reply to #25

Re: Probability Help

05/27/2021 8:39 PM

Here, you can borrow mine....

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#7
In reply to #4

Re: Probability Help

05/26/2021 3:41 PM

You seem to have an agenda, that bias will skew any results obtained...the proper way is to have an arms length third party run a simulation... Why can't you request to be an observer of the process? This would seem the most efficient way to resolve this issue...If there is an objection to observation for transparency, then I would get suspicious...

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#8
In reply to #7

Re: Probability Help

05/26/2021 4:34 PM

Here's a random list generator....make your items 1 - 175 and your quantity 10...run the simulation 8-9 times and you will see numbers repeated...

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#9
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Re: Probability Help

05/26/2021 4:54 PM
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#30
In reply to #9

Re: Probability Help

05/29/2021 10:30 AM

I like the Monte Carlo approach. Here is a short Matlab (or Octave) program that simulates a number of tests with 10 selected from 175. The numbers can be changed to suit. It plots a histogram and prints out the number selected 0 times, 1 time, 2 times, ...etc.

% start of program

% S stores 175 random numbers

% T stores number of times each value in S is in 10 lowest values

% Tests is number of drug tests

% N Number of students selected

clear;

close all;

Tests=8 % number of drug tests

N=10 % number of students selected

T(175)=0; % number of times for each student

for k=1:Tests

S=rand(1,175); % Create 175 random numbers

[S1,ix]=sort(S); % Sort, ix stores indices of S

for i=1:N

T(ix(i))=T(ix(i))+1; %increment number of times selected

endfor

end

hist(T,max(T)+1); % plot a histogram

grid on;

title('Multiple tests');

ylabel('Students');

V=hist(T,max(T)+1)

num=0:max(T);

lbl='';

for k=1:max(T)+1

lbl=sprintf('%s %d-%d ',lbl,num(k),V(k));

end

xlabel(sprintf('Times tested %s',lbl));

% end of program

It should work if copied and pasted into a file with a ".m" extension. (Anything on a line after "%" is a comment.) It requires Matlab or Octave to run. Octave is free and can be downloaded:

https://www.gnu.org/software/octave/download

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#10

Re: Probability Help

05/26/2021 5:15 PM

Randomness and probability concepts are used to fix results. The object of drug testing is for taxpayer funding, not drug abuse. No one cares about drug abuse. We have more dying every year, that died in 10 years of Vietnam. It's a zero taxed and zero regulated product. Many millionaires have been made on this. They have power and influence now.

Test ALL students and ALL faculty, and see the results. This eliminates and exposes the corruption. And education hypocrisy.

What's good for the goose is good for the gander. We love to test poor people, why not test government employees and professionals.....like politicians, doctors and lawyers, judges and academics? Why not juries?

It's all so phony.

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#12

Re: Probability Help

05/27/2021 4:30 AM

I used to write BASIC programs and had the occaisional need to generate a random list of numbers. I found that I was getting the same list each time and later learnt thst I needed to "seed" the list before starting the Random function. This gave me a new list each time.

I have just looked at Excel and there is a function,

=RANDARRAY([rows],[columns],[min],[max],[integer])
which can be used.
Perhaps the use of Excel could help.
[integer] is TRUE or FALSE

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#13
In reply to #12

Re: Probability Help

05/27/2021 5:45 AM

Perhaps the reason you are only doing random testing is because testing all would be too expensive? So here is a thought. Randomly pick three months out of the year to test all. That would help keep cost down, keep it fair, and still have the element of surprise.

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#34
In reply to #12

Re: Probability Help

05/31/2021 7:18 AM

Over the years there have been many times when I have "shot myself in the foot" by not having unique seeds for a random number generator. A perfectly good random number generator will produce the exact same output over and over if it starts each run with the same seed.

But, if the full "output" is not duplicated then do we have a seed issue here?

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#15

Re: Probability Help

05/27/2021 9:32 AM

Easy to do the experiment in a spreadsheet

Lots of kids selected twice, and in this second sample 73 and 153 both selected 3 times.

One word of warning: I haven't bothered to eliminate the possibility of the same kid being selected twice in any one months test.

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#17
In reply to #15

Re: Probability Help

05/27/2021 11:21 AM

ga,... there's a moral to this thread,... be careful what you ask a engineering community...

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#18

Re: Probability Help

05/27/2021 12:18 PM

My thoughts exactly, Haymaker. Test everyone or test no one.

My experience suggests everyone has an agenda. Since we are dealing with mostly children under the age of 18, privacy is necessary. That makes it tough to have a transparent process.

Like I said, I know that I cannot prove that someone is abusing policy/protocol, but man alive...

Thanks again for everyone's input. I'll let you all know how things turn out...

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#19

Re: Probability Help

05/27/2021 12:39 PM

And you are correct SolarEagle. I do have an agenda. The process is flawed and there is absolutely no oversight!

When you consider the probabilities involved, non-existent websites, and highly coincidental timing, I just have an extremely hard time putting any faith in this process at all.

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#20
In reply to #19

Re: Probability Help

05/27/2021 2:17 PM

You seem to be overly comfortable in playing the role of victim...Accusation without evidence is prejudicial on either side of the argument...You may be guilty of the very thing you are accusing others of....

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#21
In reply to #20

Re: Probability Help

05/27/2021 2:34 PM

But if he did not question then he would not learn.

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#22
In reply to #20

Re: Probability Help

05/27/2021 2:44 PM

Me? Victim? No sir. The victims are the kids that are being targeted under the auspice of random testing. The victims are the taxpaying citizens who support our "drug-free" school. I'm just a process engineer who has identified an extremely flawed process. Or perhaps a fraud process. Either way, someone ought to make a stink, hadn't they?

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#23
In reply to #22

Re: Probability Help

05/27/2021 4:46 PM

I don't think you are going to get very far walking in the door making accusations...again you have no proof of any wrong doing...You certainly are within your rights to run simulations and compare results, then present them to those that are conducting the selection process, if indeed the findings are beyond the realm of possibility...there's nothing wrong with oversight activity, in fact I would encourage it, but premature accusations will poison the process from the outset and result in nothing but bad blood...So far we have run simulations here and a student being picked 3,4 or even 5 times in the course of a year is in fact possible....In fact there is a high probability that at least one student will be picked 3 times...So the first thing you need to know is precisely how many students were picked how many times, not just what people think they remember...people with an agenda tend to exaggerate to make a point...this is just human nature...

If you want my personal opinion, I don't see any reason for drug testing children, and in athletes, I would just test the winners....but if you're going to go to a school that has these rules in place, then you must resolve yourself to follow the rules or go elsewhere...

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#27
In reply to #23

Re: Probability Help

05/27/2021 9:45 PM

...."Is random drug testing of students legal? In June 2002, the U.S. Supreme Court broadened the authority of public schools to test students for illegal drugs. The court ruled to allow random drug tests for all middle and high school students participating in competitive extracurricular activities."...May 26, 2020

.."How accurate are drug tests? Is there a possibility a test could give a false positive?

The accuracy of drug tests from a certified lab is very high, and confirmation tests can help to rule out any false positives. Usually, samples are divided so that if an initial test is positive, a confirmation test can be conducted. Federal guidelines are in place to ensure accuracy and fairness in drug-testing programs."...

https://www.silverpinestreatmentcenter.com/addiction-blog/what-can-cause-false-positives-for-drug-test/

https://www.drugabuse.gov/drug-topics/drug-testing/frequently-asked-questions-about-drug-testing-in-schools

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#24
In reply to #22

Re: Probability Help

05/27/2021 7:44 PM

Every test has a false negative and false positive rate. Don't eat poppy seed muffins for breakfast.

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#33
In reply to #24

Re: Probability Help

05/31/2021 7:10 AM

Poppy seed bagels, toasted with warm butter. Now that is a threat to clean test results.

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#28

Re: Probability Help

05/28/2021 4:58 PM

Perhaps a math instructor at the school could have some "believability" if he/she were to state the probability of these results. Random number generators are a common function within computerized math or scientific math calculations. They are generally NOT fully random and can even give predicable results if they are poorly written or are given a "seed" number that is constant each time they are run. You have enough information to entertain a good high-school math instructor and probably enough to have the parents require a better drug test protocol.

--JMM

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#31

Re: Probability Help

05/30/2021 5:32 AM

I understand your schepticism and support the need to question the process.

As others have already commented, some "random" number generators will produce exceptionally similar patterns if started "from scratch" and not somehow disrupted with some "seed" number.

From what you have described, this would seem to be the case in this instance.

You might like to suggest to the administrator to do a single run for the complete year and then use that to select 1 to 10, 11 to 20, 21 to 30 and so on through the year. The random number generator and the original list sequence are independent so there should then be significantly less bias in the results, but still a simple process.

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#32
In reply to #31

Re: Probability Help

05/30/2021 6:59 AM

I liked that idea, but, then I realised that if someone gets tested in one of the first sessions, they then know that they're free to use drugs for the rest of the year.

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#35

Re: Probability Help

05/31/2021 3:28 PM

A number of CR4 members have provided the math that randomly being selected 3 times is quite possible. With the original posting listing "as many as 4-5 times during the same school year" the odds might be longer than the "3 times" that has been used for the math examples.

It would be interesting to have exact numbers for a school year posted.

Are all the tests actually suppose to be random? I have known employers that have tested the person or persons that they suspect of drug abuse in a group with randomly selected people to make the whole test "random".

It has been a long time since I was in High School. Most of the drugs being abused today didn't even exist back then. It makes me feel kind of old to admit this but I don't recall for sure if drug tests even existed back then. But I do recall several cases of "my little Jimmy would never do that" when little Jimmy was doing it. There were also cases of "he's probably a stoner" when I don't think he was. Is there any chance that school admin has intel on little Jimmy (or his close friends, older brother, etc.) and that is why he is "randomly" selected 4 or 5 times in a school year?

A small comment of defense for the OP. I know of a company where 100% of the "random" people selected were people that the small company's owner was not happy with that month. There is a lasting harm done to the relationship between ruler and ruled when ruler uses "random" drug tests as a weapon against the ruled. The OP is a coach and therefore is on the receiving end of grief that he is not responsible for.

Are the parents that are vocal against the multiple tests also vocal against other things and therefore owning a history of causing grief for the people with the power to select the "random" test subjects?

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#36
In reply to #35

Re: Probability Help

06/01/2021 8:55 PM

Simplistically, with any single 1 ''pick'' of 10 ''draws'' done in 9 ''cycles'' per school year, each student has:

( 1 student-per-pick ) x ( 10 picks-per-draw ) x ( 9 cycles-per-year = 90 )

=> ninety chances in 135 picks per year, which is exactly, 2-out-of-3 chances of being picked, at least once.

So, a few won't be picked at all, and a few others will be picked at least twice, but FIVE times is planely rather ''long odds''...

(I have to add that, once a particular student IS picked, why does THAT student get returned to the ''pool'' ???)

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#37
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Re: Probability Help

06/02/2021 5:51 AM

"(I have to add that, once a particular student IS picked, why does THAT student get returned to the ''pool'' ???)"

See my answer #32 above

https://cr4.globalspec.com/comment/1309874/Re-Probability-Help

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#38
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Re: Probability Help

06/02/2021 6:29 AM

Winning the lottery is pretty long odds, but it happens every day...sometimes twice...

"(CNN)Two days. Two back-to-back lottery wins. Both in the six figures.

It must feel great to be Orlene Peterson right about now. The Idaho woman won a $300,000 lottery prize -- and then the very next day, snagged $200,000 in another win. What are the odds? Well, officials at the Idaho Lottery crunched the numbers. And they estimate it's about 1 in 282.5 million. Peterson's stroke of good luck began late January when she bought some lottery tickets in Hayden in northern Idaho."..

https://www.cnn.com/2021/02/08/us/idaho-lottery-woman-wins-twice-trnd/index.html

https://theuijunkie.com/bill-morgan-lottery/

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#39
In reply to #38

Re: Probability Help

06/02/2021 7:01 AM

Perhaps the odds of winning the lottery twice were 1 in 282.5 million, but what would the odds be of it happening every year?

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#40
In reply to #38

Re: Probability Help

06/02/2021 10:09 AM

Lucky? Depends on how the past lottery winners turned out. Especially if they don’t know how to handle the windfall.

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#41
In reply to #36

Re: Probability Help

06/02/2021 3:25 PM

Watching my units more carefully,

I must correct myself by re-stating as ''( 9 draws-per-year = 90 )''

and, say,

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''illigitimi non carborundum...''(i.e.: don't let the fatherless (self-deluding,sabotaging, long-term-memory-impaired, knee-jerking, cheap-shotting, mono-syllabic, self-annointed, shadow-lurking, back-biting, off-topic-inquisitors) grind you down...)
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