Probability Help

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.

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

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

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)!)p^kq^n-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

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.

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) ! ) ) p^k q^n-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.

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 ...

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.

Shouldn't that be p = 10/175?

Shouldn't that be p = 10/175?

Yup, I need to clean my glasses...

Here, you can borrow mine....

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...

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...

I ran the simulation about 7 times and got three 88's...

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.

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

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.

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?

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.

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

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...

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.

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....

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

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?

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...

...."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

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

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

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

Cheap and easy....

https://www.amazon.com/Natural-Bingo-Engraved-Perfect-Replacements/dp/B07N95QJ22

https://www.amazon.com/Table-Top-Toys-Cage-Deluxe-Brushed/dp/B0732PH492/ref=pd_bxgy_3/138-7852251-4244908?_encoding=UTF8&pd_rd_i=B0732PH492&pd_rd_r=2cff805d-f784-42d3-8e14-ac6dd75b6fc9&pd_rd_w=tndT5&pd_rd_wg=dfY8F&pf_rd_p=fd3ebcd0-c1a2-44cf-aba2-bbf4810b3732&pf_rd_r=3C0382S451K2H08W3QC9&psc=1&refRID=3C0382S451K2H08W3QC9

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.

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.

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?

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'' ???)

"(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

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/

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?

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

Watching my units more carefully,

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

and, say,