Question about random test




New guy here. I'm a union Rep. I help represent over 8300 members. Company policy and federal mandate says that each member on average will be drug tested once a year. That's 8300 tests in one year.

I'm representing a guy that has been drug tested once a quarter for the last 6 quarters. He has no history of drug use and is not on any substance abuse profile.

Before I jump into this issue head first, is like to have numbers to back me up. What is the probability of him getting a random test 6 quarters in a row.

Thanks for the help


I'm going to make some simplistic assumptions here, namely

  1. All members have an equal chance of being selected each time a test is run (as opposed to randomly going down the list one by one until the entire list is exhausted)
  2. The 8,300 tests performed a year are spaced out evenly over time. So we might suppose that 2,075 tests are performed each quarter
The probability of being selected for a single test is 1/8,300. Conversely, the probability of NOT being selected for a single test is 1 - 1/8,300 = 8,299/8,300.

The probability of NEVER being selected when 2,075 tests are run is

Conversely, the probability of being selected at least once when 2,075 tests are run is
1 - (8,299/8,300)^2,075 = 0.22​

So in a single quarter, there's about a 22% chance that a particular person will get selected for a drug test. Now if we consider 6 consecutive quarters (again assuming that the selections are completely independent each time), then the probability that the same person gets selected at least once each quarter is
0.22^6 = 0.0001​

I would ask this person for their lottery numbers :cool: